Converse, Inverse, and Contrapositive
Hi, and welcome to this video on mathematical statements! Today, we’ll be exploring the logic that appears in the language of math. Specifically, we will learn how to interpret a math statement to create what are known as converse, inverse, and contrapositive statements. These, along with some reasoning skills, allow us to make sense of problems presented in math. Let’s get started!
Let’s first take a look at a basic statement , which can be either true or false, but never both. For example, a declarative statement pronounces a fact, like “the Sun is hot.” We know this is a statement because the Sun cannot be both hot and not hot at the same time. This declarative statement could also be referred to as a proposition .
Two independent statements can be related to each other in a logic structure called a conditional statement . The first statement is presented with “if,” and is referred to as the hypothesis . The second statement is linked with “then”, and is known as the conclusion . The notation associated with conditional statements typically uses the variable \(p\) for the hypothesis statement, and \(q\) for the conclusion.
In words, this would be read as, “If \(p\), then \(q\).”
When the hypothesis and conclusion are identified in a statement, three other statements can be derived:
- The contrapositive statement is a combination of the previous two. The positions of \(p\) and \(q\) of the original statement are switched, and then the opposite of each is considered: \(\sim q \rightarrow \sim p\) (if not \(q\), then not \(p\)).
An example will help to make sense of this new terminology and notation. Let’s start with a conditional statement and turn it into our three other statements.
The first step is to identify the hypothesis and conclusion statements. Conditional statements make this pretty easy, as the hypothesis follows if and the conclusion follows then . The hypothesis is it is raining and the conclusion is grass is wet .
Now we can use the definitions that we introduced earlier to create the three other statements.
- Our contrapositive statement would be: “If the grass is NOT wet, then it is NOT raining.”
You may be wondering why we would want to go through the trouble of rearranging and considering the “opposite” of the hypothesis and conclusion statements. How is this helpful? The key is in the relationship between the statements. If we know that a statement is true (or false), then we can assume that another is also true (or false). The statements that are related in this way are considered logically equivalent .
For example, consider the statement, “If it is raining, then the grass is wet” to be TRUE. Then you can assume that the contrapositive statement, “If the grass is NOT wet, then it is NOT raining” is also TRUE.
Likewise, the converse statement, “If the grass is wet, then it is raining” is logically equivalent to the inverse statement, “If it is NOT raining, then the grass is NOT wet.”
These relationships are particularly helpful in math courses when you are asked to prove theorems based on definitions that are already known. Much of that work is beyond the scope of this video, but the following examples will help to illustrate the relationships of logically equivalent statements.
Here is a typical example of a TRUE statement that would be made in a geometry class based on the definition of congruent angles :
As you can see, this is not a conditional statement, but we can rewrite it in the “if-then” structure to identify the hypothesis and conclusion statements as follows:
Now we have a hypothesis and a conclusion.
Because the conditional statement and the contrapositive are logically equivalent, we can assume the following to be TRUE:
It follows that the converse statement, “If two angles are congruent, then the two angles have the same measure,” is logically equivalent to the inverse statement, “If two angles do NOT have the same measure, then they are NOT congruent.”
Here is another example of a TRUE statement:
The conditional statement would be “If a figure is a square, then it is a rectangle,” which gives us our hypothesis and conclusion.
Because the contrapositive statement is logically equivalent, we can assume that “If the figure is NOT a rectangle, then the figure is NOT a square” is also a TRUE statement.
However, the converse statement can be disproved.
As can be seen in the diagram above, squares are a type of rectangle and a rectangle is a type of polygon . However, a square is a special type of rectangle that has four sides of equal length. Not all rectangles have four equal sides like a square, so our converse statement is FALSE.
Accordingly, the inverse statement is also FALSE because they are logically equivalent:
In summary, the original statement is logically equivalent to the contrapositive, and the converse statement is logically equivalent to the inverse.
That is a lot to take in! Let’s end this video with an example for you to process how to analyze a statement to write the converse, inverse, and contrapositive statements.
For this exercise, don’t worry about whether the statements are true or false. The statement is:
Now, pause the video and see if you can figure out the converse, inverse, and contrapositive statements. Remember, it helps to first turn our original statement into a conditional statement so you know the hypothesis and conclusion.
Okay, let’s see if you figured it out!
The conditional statement would be: “If all figures are four-sided planes, then figures are rectangles.” This gave us our hypothesis and conclusion.
Here are the converse, inverse, and contrapositive statements based on the hypothesis and conclusion:
Converse : “If figures are rectangles, then figures are all four-sided planes.”
Inverse : “If figures are NOT all four-sided planes, then they are NOT rectangles.”
Contrapositive : “If figures are NOT rectangles, then the figures are NOT all four-sided planes.”
That’s all for this review! Thanks for watching, and happy studying!
Converse, Inverse, and Contrapositive Practice Questions
Which statement is NOT considered a conditional statement?
If you mow the lawn, then I will pay you for your hard work.
If you pay your power bill, then you will have electricity.
If you do not buy firewood, then you will be cold.
The sun is shining, because it is summer.
Conditional statements are also considered “If-Then” statements. An “If-Then” statement consists of a hypothesis (if) and a conclusion (then). For example, If it is snowing, then it is cold. The logic structure of conditional statements is helpful for deriving converse, inverse, and contrapositive statements.
What is the inverse statement of the following conditional statement? If it is snowing, then it is cold.
If it is not snowing, then it is cold.
If it is not snowing, then it is not cold.
If it is cold, then it might be snowing.
If it is cold, then it is not warm.
An inverse statement assumes the opposite of each of the original statements. The opposite of “If it is snowing” would be “If it is not snowing.” The opposite of “then it is cold” would be “then it is not cold.”
What is the contrapositive statement for the following conditional statement? If it is a triangle, then it is a polygon.
If it is not a triangle, then it is not a polygon.
If it is not a polygon, then it is not a triangle.
If it is a triangle, then it is a polygon.
If it is a polygon, it is a triangle.
A contrapositive statement occurs when you switch the hypothesis and the conclusion in a statement, and negate both statements. In this example, when we switch the hypothesis and the conclusion, and negate both, the result is: If it is not a polygon, then it is not a triangle.
Identify p (hypothesis) and q (conclusion) in the following conditional statement. If a figure is a triangle, then it has three angles.
q : If a figure is a triangle p : Then it has congruent angles
p : A figure is a polygon q : Can have three angles
p : If a figure is not a triangle q : Then it does not have three angles
p : If a figure is a triangle q : Then it has three angles
The hypothesis ( p ) of a conditional statement is the “if” portion. The conclusion ( q ) of a conditional statement is the “then” portion.
Which item shows the math statements matched with the correct logic symbols?
Conditional Statement: q → p Converse: q → p Inverse: ~ p → ~ q Contrapositive: ~ p → ~ q
Conditional Statement: p → q Converse: q → p Inverse: ~ p → ~ q Contrapositive: ~ q → ~ p
Conditional Statement: r → t Converse: q → p Inverse: ~ t → ~ r Contrapositive: ~ q → ~ p
Conditional Statement: p → q Converse: q → p Inverse: p → q Contrapositive: ~ h → ~ c
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by Mometrix Test Preparation | Last Updated: August 30, 2024
Converse, Inverse & Contrapositive of Conditional Statement
Converse, inverse, and contrapositive of a conditional statement.
What we want to achieve in this lesson is to be familiar with the fundamental rules on how to convert or rewrite a conditional statement into its converse, inverse, and contrapositive.
But first, we need to review what a conditional statement is because it is the foundation or precursor of the three related sentences that we are going to discuss in this lesson.
What is a Conditional Statement?
A conditional statement takes the form “If [latex]p[/latex], then [latex]q[/latex]” where [latex]p[/latex] is the hypothesis while [latex]q[/latex] is the conclusion. A conditional statement is also known as an implication .
Sometimes you may encounter (from other textbooks or resources) the words “antecedent” for the hypothesis and “consequent” for the conclusion. Don’t worry, they mean the same thing.
In addition, the statement “If [latex]p[/latex], then [latex]q[/latex]” is commonly written as the statement “[latex]p[/latex] implies [latex]q[/latex]” which is expressed symbolically as [latex]{\color{blue}p} \to {\color{red}q}[/latex].
Given a conditional statement, we can create related sentences namely: converse , inverse , and contrapositive . They are related sentences because they are all based on the original conditional statement.
Let’s go over each one of them!
The Converse of a Conditional Statement
For a given conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we can write the converse statement by interchanging or swapping the roles of the hypothesis and conclusion of the original conditional statement. Therefore, the converse is the implication [latex]{\color{red}q} \to {\color{blue}p}[/latex].
Notice, the hypothesis [latex]\large{\color{blue}p}[/latex] of the conditional statement becomes the conclusion of the converse. On the other hand, the conclusion of the conditional statement [latex]\large{\color{red}p}[/latex] becomes the hypothesis of the converse.
The Inverse of a Conditional Statement
When you’re given a conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], the inverse statement is created by negating both the hypothesis and conclusion of the original conditional statement. Thus, the inverse is the implication ~[latex]\color{blue}p[/latex] [latex]\to[/latex] ~[latex]\color{red}q[/latex].
The symbol ~[latex]\color{blue}p[/latex] is read as “not [latex]p[/latex]” while ~[latex]\color{red}q[/latex] is read as “not [latex]q[/latex]” .
The Contrapositive of a Conditional Statement
Suppose you have the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement.
In other words, to find the contrapositive, we first find the inverse of the given conditional statement then swap the roles of the hypothesis and conclusion. Therefore, the contrapositive of the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex] is the implication ~[latex]\color{red}q[/latex] [latex]\to[/latex] ~[latex]\color{blue}p[/latex].
Truth Tables of a Conditional Statement, and its Converse, Inverse, and Contrapositive
Now that we know how to symbolically write the converse, inverse, and contrapositive of a given conditional statement, it is time to state some interesting facts about these logical statements.
To save time, I have combined all the truth tables of a conditional statement, and its converse, inverse, and contrapositive into a single table .
Here are some of the important findings regarding the table above:
- The conditional statement is NOT logically equivalent to its converse and inverse.
- The conditional statement is logically equivalent to its contrapositive. Thus, [latex]{\color{blue}p} \to {\color{red}q}[/latex] [latex] \equiv [/latex] ~[latex]\color{red}q[/latex] [latex]\to[/latex] ~[latex]\color{blue}p[/latex].
- The converse is logically equivalent to the inverse of the original conditional statement. Therefore, [latex]{\color{red}q} \to {\color{blue}p}[/latex] [latex] \equiv [/latex] ~[latex]\color{blue}p[/latex] [latex]\to[/latex] ~[latex]\color{red}q[/latex].
You might also like these tutorials:
- Introduction to Truth Tables, Statements, and Logical Connectives
- Truth Tables of Five (5) Common Logical Connectives or Operators
Become a math whiz with AI Tutoring, Practice Questions & more.
Converse, Inverse, Contrapositive
Given an if-then statement "if p , then q ," we can create three related statements:
A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. For instance, “If it rains, then they cancel school.” "It rains" is the hypothesis. "They cancel school" is the conclusion.
To form the converse of the conditional statement, interchange the hypothesis and the conclusion. The converse of "If it rains, then they cancel school" is "If they cancel school, then it rains."
To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.”
To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement. The contrapositive of "If it rains, then they cancel school" is "If they do not cancel school, then it does not rain."
| Statement | If , then . |
| Converse | If , then . |
| Inverse | If not , then not . |
| Contrapositive | If not , then not . |
If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.
| Statement | If two angles are congruent, then they have the same measure. |
| Converse | If two angles have the same measure, then they are congruent. |
| Inverse | If two angles are not congruent, then they do not have the same measure. |
| Contrapositive | If two angles do not have the same measure, then they are not congruent. |
In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!
| Statement | If a quadrilateral is a rectangle, then it has two pairs of parallel sides. |
| Converse | If a quadrilateral has two pairs of parallel sides, then it is a rectangle. |
| Inverse | If a quadrilateral is not a rectangle, then it does not have two pairs of parallel sides. |
| Contrapositive | If a quadrilateral does not have two pairs of parallel sides, then it is not a rectangle. |
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Contrapositive and Converse
You may come across different types of statements in mathematical reasoning where some are mathematically acceptable statements and some are not acceptable mathematically. In mathematics, we observe many statements with “if-then” frequently. For example, consider the statement. Contrapositive and converse are specific separate statements composed from a given statement with “if-then”. Before getting into the contrapositive and converse statements, let us recall what are conditional statements. A conditional statement is formed by “if-then” such that it contains two parts namely hypothesis and conclusion. Hypothesis exists in the”if” clause, whereas the conclusion exists in the “then” clause.
What are Contrapositive Statements?
It is easy to understand how to form a contrapositive statement when one knows about the inverse statement. To create the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. First, form the inverse statement, then interchange the hypothesis and the conclusion to write the conditional statement’s contrapositive.
Click here to know how to write the negation of a statement .
In other words, contrapositive statements can be obtained by adding “not” to both component statements and changing the order for the given conditional statements.
What are Converse Statements?
The converse statements are formed by interchanging the hypothesis and conclusion of given conditional statements.
Thus, we can relate the contrapositive, converse and inverse statements in such a way that the contrapositive is the inverse of a converse statement.
This can be better understood with the help of an example.
Example: Consider the following conditional statement.
If a number is a multiple of 8, then the number is a multiple of 4.
Write the contrapositive and converse of the statement.
Given conditional statement is:
The converse of the above statement is:
If a number is a multiple of 4, then the number is a multiple of 8.
The inverse of the given statement is obtained by taking the negation of components of the statement.
If a number is not a multiple of 8, then the number is not a multiple of 4.
Now, the contrapositive statement is:
If a number is not a multiple of 4, then the number is not a multiple of 8.
All these statements may or may not be true in all the cases. That means, any of these statements could be mathematically incorrect.
Contrapositive vs Converse
The differences between Contrapositive and Converse statements are tabulated below.
| Suppose “if p, then q” is the given conditional statement “if ∼q, then ∼p” is its contrapositive statement. Note: ∼ represents the negation or inverse statement | Suppose “if p, then q” is the given conditional statement “if q, then p” is its converse statement. |
| Example: The contrapositive statement for “If a number n is even, then n is even” is “If n is not even, then n is not even. | Example: The converse statement for “If a number n is even, then n is even” is “If a number n is even, then n is even. |
| Mathematical representation: Conditional statement: p ⇒ q Contrapositive statement: ~q ⇒ ~p | Mathematical representation: Conditional statement: p ⇒ q Converse statement: q ⇒ p |
We can also construct a truth table for contrapositive and converse statement.
The truth table for Contrapositive of the conditional statement “If p, then q” is given below:
| p | q | ~p | ~q | ~q ⇒ ~p |
| T | T | F | F | T |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | T | T |
Similarly, the truth table for the converse of the conditional statement “If p, then q” is given as:
| p | q | q ⇒ p |
| T | T | T |
| T | F | T |
| F | T | F |
| F | F | T |
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What Are the Converse, Contrapositive, and Inverse?
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Conditional statements make appearances everywhere. In mathematics or elsewhere, it does not take long to run into something of the form “If P then Q .” Conditional statements are indeed important. What is also important are statements related to the original conditional statement by changing the position of P , Q, and the negation of a statement. Starting with an original statement, we end up with three new conditional statements that are named the converse, the contrapositive, and the inverse .
Before we define the converse, contrapositive, and inverse of a conditional statement, we need to examine the topic of negation. Every statement in logic is either true or false. The negation of a statement simply involves the insertion of the word “not” at the proper part of the statement. The addition of the word “not” is done so that it changes the truth status of the statement.
Looking at an example helps. The statement “The right triangle is equilateral” has negation “The right triangle is not equilateral.” The negation of “10 is an even number” is the statement “10 is not an even number.” Of course, for this last example, we could use the definition of an odd number and instead say that “10 is an odd number.” We note that the truth of a statement is the opposite of that of the negation.
Examine this idea in a more abstract setting. When the statement P is true, the statement “not P ” is false. Similarly, if P is false, its negation “not P ” is true. Negations are commonly denoted with a tilde ~. So instead of writing “not P ” we can write ~ P .
Converse, Contrapositive, and Inverse
Now we can define the converse, contrapositive, and inverse of a conditional statement. We start with the conditional statement “If P then Q .”
- The converse of the conditional statement is “If Q then P .”
- The contrapositive of the conditional statement is “If not Q then not P .”
- The inverse of the conditional statement is “If not P then not Q .”
We will see how these statements work with an example. Suppose we start with the conditional statement “If it rained last night, then the sidewalk is wet.”
- The converse of the conditional statement is “If the sidewalk is wet, then it rained last night.”
- The contrapositive of the conditional statement is “If the sidewalk is not wet, then it did not rain last night.”
- The inverse of the conditional statement is “If it did not rain last night, then the sidewalk is not wet.”
Logical Equivalence
We may wonder why it is important to form these other conditional statements from our initial one. A careful look at the above example reveals something. Suppose that the original statement “If it rained last night, then the sidewalk is wet” is true. Which of the other statements have to be true as well?
- The converse “If the sidewalk is wet, then it rained last night” is not necessarily true. The sidewalk could be wet for other reasons.
- The inverse “If it did not rain last night, then the sidewalk is not wet” is not necessarily true. Again, just because it did not rain does not mean that the sidewalk is not wet.
- The contrapositive “If the sidewalk is not wet, then it did not rain last night” is a true statement.
What we see from this example (and what can be proved mathematically) is that a conditional statement has the same truth value as its contrapositive. We say that these two statements are logically equivalent. We also see that a conditional statement is not logically equivalent to its converse and inverse.
Since a conditional statement and its contrapositive are logically equivalent, we can use this to our advantage when we prove mathematical theorems. Rather than prove the truth of a conditional statement directly, we can instead use the indirect proof strategy of proving the truth of that statement’s contrapositive. Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true.
It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement , they are logically equivalent to one another. There is an easy explanation for this. We start with the conditional statement “If Q then P ”. The contrapositive of this statement is “If not P then not Q .” Since the inverse is the contrapositive of the converse, the converse and inverse are logically equivalent.
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Converse, Inverse, and Contrapositive
As your English teacher would say, good writers vary their sentence structure. The same is true of conditional statements: after a while, the If-Then formula becomes a real snoozefest. Some ways to mix it up are: "All things satisfying hypothesis are conclusion " and " Conclusion whenever hypothesis ."
However, mathematicians can be drier than the Sahara desert: they tend to write conditional statements as a formula p → q , where p is the hypothesis and q the conclusion. In fact, the old saying, "Mind your p 's and q 's," has its origins in this sort of mathematical logic.
Sample Problem
Identify p and q in the following statements, translating them into p → q form.
(A) If it rains outside, then flowers will grow tomorrow. (B) I cut off a finger whenever I peel rutabagas. (C) All dogs go to heaven.
For (A), p = "it rains outside" and q = "flowers will grow tomorrow."
In (B), we may rewrite the statement as "If I peel rutabagas, then I cut off a finger," telling us that p = "I peel rutabagas" and q = "I cut off a finger."
Finally, we may rewrite (C) as "If it is a dog, then it will go to heaven," yielding p = "it is a dog" and q = "it will go to heaven."
The hypothesis and conclusion play very different roles in conditional statements. Duh. In other words, p → q and q → p mean very different things. It's kind of like subtraction: 5 – 3 gives a different answer than 3 – 5. To highlight this distinction, mathematicians have given a special name to the statement q → p : it's called the converse of p → q .
No, not those Converse.
Write the converse of the statement, "If something is a watermelon, then it has seeds."
We want to switch the hypothesis and the conclusion, which will give us: "If something has seeds, then it is a watermelon." Of course, this converse is obviously false, since apples, cucumbers, and sunflowers all have seeds and are not watermelons. At least not during their day jobs.
There are some other special ways of modifying implications. For example, if you negate (that means stick a "not" in front of) both the hypothesis and conclusion, you get the inverse : in symbols, not p → not q is the inverse of p → q . Sometimes mathematicians like to be even more brief than this, so they'll abbreviate "not" with the symbol "~". So we can also write the inverse of p → q as ~ p → ~ q .
Finally, if you negate everything and flip p and q (taking the inverse of the converse, if you're fond of wordplay) then you get the contrapositive . Again in symbols, the contrapositive of p → q is the statement not q → not p , or ~ q → ~ p . Fancy.
What is the inverse of the statement "All mirrors are shiny?" What is its contrapositive?
If we abbreviate the first statement as mirror → shiny, then the inverse would be not mirror → not shiny and the contrapositive would be not shiny → not mirror. Written in English, the inverse is, "If it is not a mirror, then it is not shiny," while the contrapositive is, "If it is not shiny, then it is not a mirror."
While we've seen that it's possible for a statement to be true while its converse is false, it turns out that the contrapositive is better behaved. Whenever a conditional statement is true, its contrapositive is also true and vice versa. Similarly, a statement's converse and its inverse are always either both true or both false. (Note that the inverse is the contrapositive of the converse. Can you show that?)
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CONVERSE INVERSE AND CONTRAPOSTIVE
Generally the conditional if p then q is the connective most often used in reasoning.
However; with some changes in words in the original statement, additional conditionals can be formed. These new conditionals are called the inverse, the converse, and the contrapositive.
Definition of inverse :
Inverse is a statement formed by negating the hypothesis and conclusion of the original conditional. Symbolically, the inverse is written as (~p ⇒ ~q)
Right angle is defined as- an angle whose measure is 90 degrees. If you are to write it as inverse statement, it can be done like: If an angle is not a right angle, then it does not measure 90.
Definition of converse :
Converse is a statement formed by interchanging the hypothesis and the conclusion i.e. original conditional (p ⇒ q) is written as (q ⇒ p)
"If two lines don't intersect, then they are parallel", it can be written as "If two lines are parallel, then they don't intersect."
Note : a conditional (p ⇒ q) and its converse (q ⇒ p) may or may not be true. It is important that the truth value of each converse is judged on its own merits.
Definition of contrapositive :
Contrapositive is a statement formed by negating both the hypothesis and conclusion (p q) and also then interchanging these negations (~ q ⇒ ~p).
the statement ‘A triangle is a threesided polygon’ is true; its contrapositive, ‘A polygon with greater or less than three sides is not a triangle’ is true too.
Remember: a conditional (p ⇒ q) and its contrapositive (~ q ⇒ ~p) must have the same truth value. When a conditional is true, it's contrapositive is also true and when a conditional is false, it's contrapositive is also false.
Problem 1 :
What is the inverse of the statement “If two triangles are not similar, their corresponding angles are not congruent”?
Considering the original statements as p and q, its inverse statement can be written in the form (~p ⇒ ~q).
So, the inverse statement is,
If two triangles are similar, their corresponding angles are congruent.
Problem 2 :
What is the inverse of the statement “If it is sunny, I will play baseball”?
So, the answer is
If it is not sunny, I will not play baseball.
Problem 3 :
What is the inverse of the statement “If Mike did his homework, then he will pass this test”?
If Mike did not do his homework, then he will not pass this test.
Problem 4 :
What is the inverse of the statement “If Julie works hard, then she succeeds”?
If Julie does not work hard, then she does not succeed.
Problem 5 :
What is the inverse of the statement “If I do not buy a ticket, then I do not go to the concert”?
If I buy a ticket, then I go to the concert.
Problem 6 :
Which statement is the inverse of "If the waves are small, I do not go surfing"?
If the waves are not small, I go surfing.
Problem 7 :
Which statement is the inverse of “If x + 3 = 7, then x = 4”?
1) If x = 4, then x + 3 = 7. 2) If x ≠ 4, then x + 3 ≠ 7.
3) If x + 3 ≠ 7, then x ≠ 4. 4) If x + 3 = 7, then x ≠ 4.
If x + 3 ≠ 7, then x ≠ 4 is the inverse of the given statement.
Problem 8 :
What is the converse of the statement “If it is sunny, I will go swimming”?
Considering the original statements as p and q, (p ⇒ q) its converse statement is written as (q ⇒ p)
So, the answer is,
If I go swimming, it is sunny.
Problem 9 :
Which statement is the converse of “If it is a 300 ZX, then it is a car”?
If it is a car, then it is a 300 ZX
Problem 10 :
What is the converse of the statement "If it is Sunday, then I do not go to school"?
If I do not go to school, then it is Sunday.
Problem 11 :
What is the converse of the statement "If Alicia goes to Albany, then Ben goes to Buffalo"?
If Ben goes to Buffalo, then Alicia goes to Albany.
Problem 12 :
What is the converse of the statement "If the Sun rises in the east, then it sets in the west"?
If the Sun sets in the west, then it rises in the east.
Problem 13 :
What is the converse of the statement "If x is an even integer, then (x + 1) is an odd integer"?
If (x + 1) is an odd integer, then x is an even integer.
Problem 14 :
What is the contrapositive of the statement, “If I am tall, then I will bump my head”?
Considering the given statements as p and q, its contrapositive statement is written in the form (~ q ⇒ ~p).
If I do not bump my head, then I am not tall.
Problem 15 :
What is the contrapositive of the statement “If I study, then I pass the test”?
If I do not pass the test, then I do not study
Problem 16 :
Given the statement, "If a number has exactly two factors, it is a prime number," what is the contrapositive of this statement?
If a number is not a prime number, then it does not have exactly two factors.
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Converse Statement
Converse Statement is a type of conditional statement where the hypothesis (or antecedent) and conclusion (or consequence) are reversed relative to a given conditional statement.
For instance, consider the statement: “If a triangle ABC is an equilateral triangle, then all its interior angles are equal.” The converse of this statement would be: “If all the interior angles of triangle ABC are equal, then it is an equilateral triangle”
In this article, we will discuss all the things related to the Converse statement in detail.
Table of Content
What is a Converse Statement?
How to write a converse statement, examples of converse statements.
- Truth Value of a Converse Statement
Truth Table for Converse Statement
- Other Types of Statements
A converse statement is a proposition formed by interchanging the hypothesis and conclusion of a conditional statement .
In simpler terms, it’s like flipping the order of “if” and “then” in a statement. For example, in the conditional statement “If it is raining, then the ground is wet”, the converse statement would be “If the ground is wet, then it is raining.”
Note: T he truth of the original statement doesn’t necessarily imply the truth of its converse, and vice versa.
Definition of Converse Statement
A converse statement is formed by exchanging the hypothesis and conclusion of a conditional statement while retaining the same meaning.
For instance, if the original statement is “If A, then B,” the converse is “If B, then A.” The validity of a converse statement doesn’t guarantee the truth of the original statement, and vice versa.
To write a converse statement, you simply switch the hypothesis and conclusion of a conditional statement while maintaining the same meaning. For example, if the original statement is “If it is raining (hypothesis), then the ground is wet (conclusion),” the converse statement would be “If the ground is wet (hypothesis), then it is raining (conclusion).” Remember, the converse statement may not always be true, even if the original statement is.
Some examples of converse statements are:
- Original Statement: If a shape is a square, then it has four equal sides. Converse Statement: If a shape has four equal sides, then it is a square.
- Original Statement: If it is summer, then the weather is hot. Converse Statement: If the weather is hot, then it is summer.
- Original Statement: If a number is divisible by 2, then it is even. Converse Statement: If a number is even, then it is divisible by 2.
- Original Statement: If a person is a teenager, then they are between 13 and 19 years old. Converse Statement: If a person is between 13 and 19 years old, then they are a teenager.
- Original Statement: If an animal is a dog, then it has fur. Converse Statement: If an animal has fur, then it is a dog.
Examples of Converse Statements in Mathematics or Logic
Some examples of converse statements in mathematics or logic:
- Original Statement: If two angles are congruent, then they have the same measure. Converse Statement: If two angles have the same measure, then they are congruent.
- Original Statement: If a number is divisible by 6, then it is divisible by 2 and 3. Converse Statement: If a number is divisible by 2 and 3, then it is divisible by 6.
- Original Statement: If two lines are perpendicular, then their slopes are negative reciprocals of each other. Converse Statement: If the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.
Converse, Inverse and Contrapositive Statements
Inverse Statement: The inverse of a conditional statement is formed by negating both the hypothesis and the conclusion of the original statement.
Contrapositive Statement: The contrapositive of a conditional statement is formed by switching the hypothesis and conclusion of the original statement and negating both.
| Statement | Converse | Inverse | Contrapositive |
|---|---|---|---|
| If p, then q | If q, then p | If not p, then not q | If not q, then not p |
Example of Inverse Statements
Original Statement: If a number is even, then it is divisible by 2. Inverse Statement : If a number is not even, then it is not divisible by 2.
Original Statement: If x > 5, then 2x > 10. Inverse Statement: If x ≤ 5, then 2x ≤ 10.
Example of Contrapositive Statements
Original Statement: If a shape is a square, then it has four equal sides. Contrapositive Statement: If a shape does not have four equal sides, then it is not a square.
Original Statement: If a number is even, then it is divisible by 2. Contrapositive: If a number is not divisible by 2, then it is not even.
To create a truth table for the converse statement, we need to consider both the original statement and its converse.
Let’s represent the original statement as “If p, then q” or “p → q” where p is the hypothesis and q is the conclusion. The converse of this statement is “If q, then p” or “q → p”. Then truth table is given by:
| Original | Converse | ||||
|---|---|---|---|---|---|
| TRUE | TRUE | FALSE | FALSE | TRUE | TRUE |
| TRUE | FALSE | FALSE | TRUE | FALSE | TRUE |
| FALSE | TRUE | TRUE | FALSE | TRUE | FALSE |
| FALSE | FALSE | TRUE | TRUE | TRUE | TRUE |
Truth Table for Inverse and Contrapositive Statement
To create a truth table for the inverse and contrapositive statements, let’s start with the original statement “If p, then q” or “p → q” where p is the hypothesis and q is the conclusion. The inverse of this statement is “If not p, then not q” or “~p → ~q”, and the contrapositive is “If not q, then not p” or “~q → ~p”. Then truth table is given by:
| Original | Inverse | Contrapositive | ||||
|---|---|---|---|---|---|---|
| TRUE | TRUE | FALSE | FALSE | TRUE | TRUE | TRUE |
| TRUE | FALSE | FALSE | TRUE | FALSE | TRUE | FALSE |
| FALSE | TRUE | TRUE | FALSE | TRUE | FALSE | TRUE |
| FALSE | FALSE | TRUE | TRUE | TRUE | TRUE | TRUE |
Solved Questions on Converse Statement
Example 1: If all squares are rectangles, are all rectangles squares?
Converse: If a shape is a rectangle, then it is a square.
The original statement says that all squares are rectangles. This is true because a square, by definition, has four sides of equal length and four right angles, making it a special type of rectangle where all sides are equal. However, the converse statement is not necessarily true. Not all rectangles are squares because rectangles can have unequal side lengths, whereas squares have all sides equal. Therefore, the converse statement is false.
Example 2: If all right angles are 90 degrees, are all 90 degree angles right angles?
Converse: If an angle measures 90 degrees, then it is a right angle.
The original statement is true because a right angle, by definition, measures 90 degrees. However, the converse statement is also true. If an angle measures 90 degrees, then it must be a right angle, as any angle measuring exactly 90 degrees forms a perfect right angle.
Example 3: If a number is divisible by 3, then it is an odd number.
Converse: If a number is an odd number, then it is divisible by 3.
The original statement is false. While it is true that all odd numbers are not divisible by 2, they are not necessarily divisible by 3. For example, the number 5 is an odd number but is not divisible by 3. Therefore, the converse statement is also false because not all odd numbers are divisible by 3.
Example 4: If a shape has four sides, then it is a quadrilateral.
Converse: If a shape is a quadrilateral, then it has four sides.
The original statement is true. A quadrilateral is defined as a polygon with four sides, so any shape with four sides is indeed a quadrilateral. Similarly, the converse statement is true. If a shape is a quadrilateral, then it must have four sides because that is a defining characteristic of a quadrilateral. Therefore, both the original statement and its converse are true.
Converse Statement: Practice Questions
Q1: If all birds have wings, do all winged creatures have beaks?
Q2: If all triangles have three sides, do all polygons with three sides have to be triangles?
Q3: If all vehicles are cars, are all cars vehicles?
Converse Statement: FAQs
What is conditional statement.
A conditional statement is a fundamental concept in logic and mathematics where a hypothesis is followed by a conclusion, often represented as “If p, then q.”
What is the Converse of a Statement?
The converse of a statement is formed by interchanging the antecedent and the consequent of a conditional statement. For example, if the original statement is “If it is raining, then the ground is wet,” the converse would be “If the ground is wet, then it is raining.”
How do Mathematicians Use Converse?
Mathematicians use the converse of a statement to explore the logical relationships between different assertions. By examining both the original statement and its converse, mathematicians can gain a deeper understanding of implications and relationships within a given context.
Is a Conditional Statement Logically Equivalent to a Converse and Inverse?
A conditional statement is not logically equivalent to its converse or inverse. While a conditional statement asserts a specific relationship between two events or conditions, the converse and inverse statements may or may not hold true in the same context.
Do the Converse and the Inverse Have The Same Truth Value?
The truth value of the converse and the inverse may differ from that of the original conditional statement. In some cases, the converse and the inverse of a true conditional statement may also be true, but this is not always the case. Each statement must be evaluated independently to determine its truth value.
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Contrapositive
Contrapositive Statements in Geometry Explained with Examples
Home › Math › Geometry › Contrapositive
What is Contrapositive?
Definition: Contrapositive is exchanging the hypothesis and conclusion of a conditional statement and negating both hypothesis and conclusion.
For example the contrapositive of “if A then B” is “if not-B then not-A”. The contrapositive of a conditional statement is a combination of the converse and inverse.
Conditional statement: A conditional statement also known as an implication. A conditional statement is in the form “If p, then q” where p is the hypothesis while q is the conclusion.
Contrapositive Statement C haracteristics
- The contrapositive of any true proposition is also true.
- Contrapositive of a true statement is also true.
- Contrapositive of a false statement is also false.
The Law of Contrapositive in Geometry
The conditional statement and its contrapositive are logically equivalent.
- Contrapositive is used when an implication has many hypotheses or when the hypothesis specifies infinitely many objects.
- Contrapositive can be used as a strong tool for proving mathematical theorems because contrapositive of a statement always has the same truth table.
Contrapositive Formula
If the conditional of a statement is p q then, we can compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement. The contrapositive of p q is q p.
The contrapositive of a conditional statement is a combination of the converse and inverse.
If the conditional statement is p q
Converse statement of p q is q p.
The hypothesis p of the conditional statement becomes the conclusion of the converse.
The conclusion q of the conditional statement becomes the hypothesis of the converse.
Inverse statement of q p is q p.
The inverse statement is obtained by negating both hypothesis and conclusion.
‘If q then p’ is a contrapositive of the conditional statement ‘if p then q’.
Contrapositive of a conditional statement is logically equivalent to its conditional statement.
Conditional Statement Examples
Conditional statement:
If it is raining, then the grass is wet. The contrapositive of this statement is: If the grass is not wet then it is not raining.
If a figure is a square then all the four sides are equal. The contrapositive of this statement is: If all the four sides are not equal then it is not a square.
If x is equal to zero, then sin(x) is equal to zero. The contrapositive of this statement is: If sin(x) is not zero, then x is not zero.
If am standing in Manchester, then I am standing in United Kingdom. The contrapositive of this statement is: If I am not standing in United Kingdom, then I am not standing in Manchester.
If a polygon is a triangle, then it has 3 sides. The contrapositive of this statement is: If the polygon does not have three sides, then it is not a triangle.
Contrapositive Truth Table example
Conditional v/s contrapositive.
p: If it rains
q: they cancel school
If the conditional statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.
| Conditional Statement | If p, then q | p q |
| Converse | If q, then p | q p |
| Inverse | If not p, then not q | p q |
| Contrapositive | If not q, then not p | q p |
The contrapositive of a conditional statement is the mixing of the converse and inverse.
“If the sun sets down” is the hypothesis
“It’s in the west” is the conclusion.
To form the converse of the conditional statement, interchange the hypothesis and the conclusion.
The converse of “The sun sets down then it’s in the west” is “The sun is in the west, if it sets down”.
To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.
The inverse of “The sun sets down then it’s in the west” is “The sun is not in the west, if it not sets down”.
To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.
The contrapositive of “The sun sets down then it’s in the west” is “The sun not sets down, if it is not in the west”.
| Statement | If two angles are congruent, then they have the same measure. (True) |
| Converse | If two angles have the same measure, then they are congruent. (True) |
| Inverse | If two angles are not congruent, then they have not the same measure. (True) |
| Contrapositive | If two angles do not have the same measure, then they are not congruent. (True) |
| Statement | If a figure is a rhombus, then its diagonals are perpendicular. (True) |
| Converse | If diagonals are perpendicular, then it is rhombus. (False) |
| Inverse | If a figure is not a rhombus, then its diagonals are not perpendicular.(False) |
| Contrapositive | If diagonals are not perpendicular, then it is not rhombus. (True) |
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Converse, Inverse, and Contrapositive of a Conditional Statement
Conditional Statement Definitions and Examples
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A conditional statement is a statement in mathematics that declares two values or sets of values to be equivalent under specific conditions. The converse, inverse, and contrapositive of a conditional statement are three ways to restate the original statement in order to better understand it. In this blog post, we will explore the Converse, Inverse, and Contrapositive of a Conditional Statement. We will look at how each one is used to restate the original statement and what benefits they offer in terms of understanding the statement.
What is a conditional statement?
A conditional statement is a statement in which one proposition is asserted to be true if and only if another proposition is also true. For example, the statement “If it rains, then the ground will be wet” is a conditional statement. The first proposition (“it rains”) is called the antecedent, while the second proposition (“the ground will be wet”) is called the consequent.
What is the converse of a conditional statement?
The converse of a conditional statement is the result of reversing the hypothesis and conclusion of the original statement. In other words, the converse of “If A, then B” is “If B, then A.” The converse is not necessarily true – it can be false or true. For example, the converse of “If it rains, then the ground is wet” would be “If the ground is wet, then it rains.” This is not always true because there are other ways for the ground to become wet (e.g., dew, sprinklers).
What is the inverse of a conditional statement?
The inverse of a conditional statement is the statement formed by reversing the hypothesis and conclusion of the original statement. For example, the inverse of the conditional statement “If it rains, then the grass will be wet” is “If the grass is not wet, then it did not rain.”
What is the contrapositive of a conditional statement?
If the contrapositive of a conditional statement is true, then the conditional statement is false. The contrapositive of a conditional statement is formed by reversing the order of the elements in the original statement and negating both the hypothesis and conclusion. For example, the contrapositive of “If it rains, then the ground will be wet” is “If the ground is not wet, then it will not rain.”
How to use the converse, inverse, and contrapositive of a conditional statement
If you know how to use the conditional statement, then you can easily understand its converse, inverse, and contrapositive. The converse of the conditional statement is “If A, then B.” The inverse of the conditional statement is “If not A, then not B.” The contrapositive of the conditional statement is “If not B, then not A.”
Here is an example:
Suppose we have a conditional statement such as “If it rains tomorrow, I will go to the movies.”
The converse of this would be “If I go to the movies tomorrow, it will rain.” The inverse would be “If it does not rain tomorrow, I will not go to the movies.” The contrapositive would be “If I do not go to the movies tomorrow, it will not rain.”
The Converse, Inverse, and Contrapositive of a Conditional Statement are all important concepts to understand when studying mathematics. The Converse of a statement is the reverse of the original statement, the Inverse is the negation of both the hypothesis and conclusion, and the Contrapositive is the inverse of the converse. These three concepts are critical to understanding mathematical proofs and solving problems.
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Converse, Inverse, and Contrapositive
This is the third post in a series on logic, with a focus on how it is expressed in English. We’ve looked at basic ideas of translating between English and logical symbols, and in particular at negation (stating the opposite). Now we are ready to consider how to change a given statement into one of three related statements.
A conditional statement and its converse
We’ll start with a question from 1999 that introduces the concepts:
Ricky has been asked to break down the statement, “A number divisible by 2 is divisible by 4,” into its component parts, and then rearrange them to find the converse of the statement. I took the question:
We commonly write such a statement symbolically as “\(p\rightarrow q\)“, where the hypothesis is p and the conclusion is q . I rewrote each part slightly to allow it to exist outside of the sentence, naming the number N to avoid needing pronouns. What was important was to rewrite the statement in if/then form.
The converse of this statement swaps the hypothesis and conclusion, making “\(q\rightarrow p\)“:
Ricky was asked to decide whether the converse is true or not, and then prove it, whichever way it goes. This part goes beyond mere logic and enters the realm of “number theory”; but commonly this sort of question is first asked in cases where the proof is not too hard, which is the case here.
To show that a statement is not always true, we only need to find an example for which it is false. In this case, an easy example is 2, or we could use 6, or 102, or whatever we like.
But the question was about the converse:
I didn’t give a proof, in part because Ricky needed to think about that for himself, but also because I didn’t know what level of proof Ricky is expected to handle. One approach is to see that any multiple of 4 can be written as 4 k for some integer k ; but that can be written as 2(2 k ), which is clearly a multiple of 2.
Converse, inverse, and contrapositive
Now we can review the meanings of all three terms, in this 1999 question, which again uses an example from basic number theory:
Doctor Kate could have asked Hollye for her answers to part A, to make sure she understands that part; but she chose to provide them:
It’s important to identify the parts of a conditional statement (if p then q ); and since two of the new statements require negations, that also might as well be done early. Notice that the negation of “is even” could have been written as “is not even”, but since every number (integer) is either odd or even, writing “is odd” is cleaner. Also, the negation of “both are even” is “at least one is not even”; this is an application of De Morgan’s law, or can be seen by considering that if it is not true that both are even, then there must be one that is not even. These ideas were discussed last time.
Now here are the new statements:
We saw the converse above; there we just swap p and q . The inverse keeps each part in place, but negates it. The contrapositive both swaps and negates the parts.
So now we know that the contrapositive, “If either m or n is odd, then m + n is odd,” is false, because there is at least one case, 3 and 7, where the hypothesis is true but the conclusion is false.
That’s the essence of a counterexample.
Doctor Kate continued, showing a way to prove that B and C (the converse and inverse) are both false. You can read that on your own, since my goal here is just to look at the logic. (We’ll have a series on proofs some time in the future.)
Rewriting the statement
Continuing, here is a similar question, where statements must first be written in conditional form:
The second statement is straightforward, but the others need thought. Doctor Achilles first defined the three forms, as we’ve already seen, and then dealt with the first case:
Thus, “all” (the universal quantifier) translates directly to a conditional. The answer, left for Hana to do, will be:
- Converse: “If x is a quadrilateral, then x is a square”; i.e. “Any quadrilateral is a square.”
- Inverse: “If x is not a square, then x is not a quadrilateral”; i.e. “Anything that is not a square is not a quadrilateral.”
- Contrapositive: “If x is not a quadrilateral, then x is not a square”; i.e. “Anything that is not a quadrilateral is not a square.”
The original statement, and the contrapositive, are true, because a square is a kind of quadrilateral; the converse and inverse are false, and a counterexample would be an oblong rectangle, which is not a square but is a quadrilateral.
The questions so far, where they dealt with truth at all, only asked about specific examples. Our last two questions will look more broadly at when these statements are equivalent.
Which can I use in a proof?
Consider this question, from 2002:
If we know a statement is true, can we conclude that the inverse is true? Doctor TWE answered with a counterexample:
Here we are using logic to talk about logic: The statement “For all p and q , \((p\rightarrow q)\rightarrow(\lnot p\rightarrow\lnot q)\)” is false! Sometimes both original and inverse are true, but we can’t conclude the latter from the former.
Giving one example where the contrapositive is true does not prove that it is always equivalent; we’ll prove it below.
In fact, the converse and inverse turn out to be equivalent to one another, though not to the original.
Why is the contrapositive equivalent?
Let’s look at one more, from 2003:
The opening statement describes the contrapositive as the inverse of the converse. What that means is this: Suppose we start with “\(p\rightarrow q\)“. Its converse is “\(q\rightarrow p\)” (swapping the order), and the inverse of that is “\(\lnot q\rightarrow\lnot p\)” (negating each part). This is the contrapositive. In the example, the converse of “If I like cats, then I have cats” is “If I have cats, then I like cats”, and the inverse of that is “If I don’t have cats, then I don’t like cats”, which is the contrapositive.
Doctor Achilles, perhaps misreading the question, answered the bigger question: Which of these are true?
In effect, he has made a truth table:
If you are unconvinced by any of the reasoning, see Why, in Logic, Does False Imply Anything? .
So the truth table for the contrapositive is that same as for the original; this is what we mean when we say that two statements are logically equivalent .
We can instead just think through the example:
Which is more convincing? That depends upon you.
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Discrete Math
Section 2.2 conditionals, converse, inverse, and contrapositive.
Many statements and theorems in mathematics are of the form “If \(X\) is true, then \(Y\) is true.”, called a conditional .
Conditionals are the basis for logical deduction, because we want to be able to make a conclusion based on known facts.
Subsection 2.2.1 Conditional (Implication)
Definition 2.2.1 ..
The conditional (or implication ), denoted by \(P \rightarrow Q\text{,}\) is a statement that is true if \(P\) if false or if \(P\) and \(Q\) are true, and is false if \(P\) is true and \(Q\) is false.
In mathematics, \(P\) is often called the hypothesis and \(Q\) is called the conclusion . In logic, \(P\) and \(Q\) are more typically called the antecedent and consequent , respectively.
It is called a conditional because the truth of the conclusion \(Q\) is conditional on the truth of the hypothesis \(P\text{.}\) It is called an implication because the truth of \(P\) implies the truth of \(Q\text{.}\)
Example 2.2.2 .
“If what you've told me is true, [then] you will have gained my trust.” (Mace Windu).
If a conditional is true, then the hypothesis \(P\) gives a condition (not necessarily the only condition) under which the conclusion \(Q\) will also be guaranteed to be true.
If the condition \(P\) is not met (i.e. is false), a conditional says nothing about the truth of the conclusion.
The truth table for a conditional is given by,
If a conditional statement is true simply because its hypothesis is false, it is said to be vacuously true (or, true by default).
Alternative forms of the conditional include, “If \(P\text{,}\) then \(Q\)”, “\(P\) implies \(Q\)”, “\(P\) is a sufficient condition for \(Q\)”, “\(P\) only if \(Q\)”, “\(Q\) if \(P\)”, and, “\(Q\) is a necessary condition for \(P\)”.
Subsection 2.2.2 Logical Connectives as Operators
These logical connectives can be thought of as operators that combine multiple statments and ouput another single statement. This is analogous to how \(+\) or \(\times\) combines two numbers and outputs a single number.
Subsection 2.2.3 Converse
Definition 2.2.3 ..
The converse , \(Q \rightarrow P\text{,}\) switches the hypothesis and conclusion.
Definition 2.2.4 .
The inverse , \(\neg P \rightarrow \neg Q\text{,}\) negates the hypothesis and conclusion.
The truth table is given by,
Subsection 2.2.4 Contrapositive
Definition 2.2.5 ..
The contrapositive , \(\neg Q \rightarrow \neg P\text{,}\) negates both statements, and reverses the order of the conditional.
In short, the contrapositive is the converse of the inverse. It has truth table given by,
Notice that the truth values for the contrapositive \(\neg Q \rightarrow \neg P\) are the same as the original condition \(P \rightarrow Q\text{.}\)
What is a Contrapositive: Understanding Logic
Logic is an essential tool in problem-solving, decision-making, and critical thinking. In the realm of logic, one concept that plays a crucial role is the contrapositive. Understanding the contrapositive allows us to analyze statements, draw conclusions, and make logical deductions. In this blog post, we will explore the definition of a contrapositive, provide examples to illustrate its application, and offer exercises to test your understanding.
Definition of a Contrapositive
A contrapositive is a statement that is formed by negating and swapping the hypothesis and the conclusion of an original conditional statement. It is a way of expressing the logical relationship between two statements. By understanding contrapositives, we can analyze the validity of arguments and identify equivalent statements.
Examples of Contrapositives
To better grasp the concept of a contrapositive, let’s consider a few examples:
- Original Statement: If a person is happy, then they smile. Contrapositive: If a person does not smile, then they are not happy.
- Original Statement: If a fruit is an apple, then it is sweet. Contrapositive: If a fruit is not sweet, then it is not an apple.
- Original Statement: If a number is even, then it is not odd. Contrapositive: If a number is odd, then it is not even.
Understanding the Contrapositive
To identify and form a contrapositive, follow these steps:
- Determine the hypothesis and the conclusion of the initial statement.
- Negate both the hypothesis and the conclusion.
- Swap the positions of the negated hypothesis and conclusion to form the contrapositive.
By analyzing the contrapositive of a statement, you can determine its logical equivalence and draw meaningful conclusions.
Identify the contrapositive in the following statements:
- If a person is a teacher, then they work at a school.
- If a substance is water, then it is wet.
- If it is a holiday, then the office is closed.
- If a book is a novel, then it is fiction.
- If a person does not work at a school, then they are not a teacher.
- If a substance is not wet, then it is not water.
- If the office is not closed, then it is not a holiday.
- If a book is not fiction, then it is not a novel.
Solving these exercises will help reinforce your understanding of contrapositives and enhance your logical reasoning skills.
Understanding the contrapositive is a valuable skill that enables us to analyze statements, draw logical conclusions, and make informed decisions. By recognizing the contrapositive of a statement, we can evaluate the validity of arguments and identify equivalent statements. Practicing exercises and quizzes related to contrapositives will further refine your logical reasoning abilities.
The critical thinking skills developed through learning about contrapositives can have a positive impact far beyond the classroom. Embrace the power of logic, and let the contrapositive guide you in unraveling the mysteries of logical thinking!
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COMMENTS
hypothesis implies that just before the current iteration begins, the elements of S sum to n. The loop replaces two copies of some number 2i with their sum 2i+1, leaving the total sum of S unchanged.Thus, when the iteration ends, the elements of S sum to n. Thus, when the algorithm halts, the elements of S are distinct powers of 2that sum to n.
The contrapositive is logically equivalent to the original statement. The converse and inverse may or may not be true. When the original statement and converse are both true then the statement is a biconditional statement. In other words, if p → q is true and q → p is true, then p ↔ q (said " p if and only if q ").
A contrapositive statement occurs when you switch the hypothesis and the conclusion in a statement, and negate both statements. In this example, when we switch the hypothesis and the conclusion, and negate both, the result is: If it is not a polygon, then it is not a triangle.
The Contrapositive of a Conditional Statement. Suppose you have the conditional statement [latex]{\color{blue}p} \to {\color{red}q}[/latex], we compose the contrapositive statement by interchanging the hypothesis and conclusion of the inverse of the same conditional statement.. In other words, to find the contrapositive, we first find the inverse of the given conditional statement then swap ...
Converse, Inverse, Contrapositive Given an if-then statement "if p , then q ," we can create three related statements: A conditional statement consists of two parts, a hypothesis in the "if" clause and a conclusion in the "then" clause. For instance, "If it rains, then they cancel school." "It rains" is the hypothesis.
What are Contrapositive Statements? It is easy to understand how to form a contrapositive statement when one knows about the inverse statement. To create the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. First, form the inverse statement, then interchange the hypothesis and the conclusion to ...
The converse of the conditional statement is "If Q then P .". The contrapositive of the conditional statement is "If not Q then not P .". The inverse of the conditional statement is "If not P then not Q .". We will see how these statements work with an example. Suppose we start with the conditional statement "If it rained last ...
The contrapositive of a statement takes the original statements, negates the hypothesis and conclusion, and swaps the order of the hypothesis and conclusion. If a statement is {eq}p \to q {/eq ...
The hypothesis and conclusion play very different roles in conditional statements. Duh. ... then you get the contrapositive. Again in symbols, the contrapositive of p → q is the statement not q → not p, or ~q → ~p. Fancy. Sample Problem. What is the inverse of the statement "All mirrors are shiny?" What is its contrapositive?
Fricasé de Pollo is a type of Cuban food. Statements 2 and 4 are logical statements; statement 1 is an opinion, and statement 3 is a fragment with no logical meaning. Four testable types of logical statements are converse, inverse, contrapositive, and counterexample statements. They can produce logical equivalence for the original statement ...
Contrapositive is a statement formed by negating both the hypothesis and conclusion (p q) and also then interchanging these negations (~ q ⇒ ~p). Example : the statement 'A triangle is a threesided polygon' is true; its contrapositive, 'A polygon with greater or less than three sides is not a triangle' is true too. Note :
Converse Statement is a type of conditional statement where the hypothesis (or antecedent) and conclusion (or consequence) are reversed relative to a given conditional statement. For instance, consider the statement: "If a triangle ABC is an equilateral triangle, then all its interior angles are equal.". The converse of this statement would ...
Contrapositive Statement: If a quadrilateral does not have 4 equal angles, then it is not a square. Conclusion A contrapositive is a form of a conditional statement. It is an outcome statement after exchanging the hypothesis and conclusion of an inverse statement, as the inverse statement is a must in calculating the contrapositive statement.
Determine the contrapositive of the conditional statement: If it is windy then the flag is moving. Step 1: Identify the hypothesis and conclusion of the conditional statement. In this case, our ...
Contents [ show] Definition: Contrapositive is exchanging the hypothesis and conclusion of a conditional statement and negating both hypothesis and conclusion. For example the contrapositive of "if A then B" is "if not-B then not-A". The contrapositive of a conditional statement is a combination of the converse and inverse.
The contrapositive of a conditional statement is formed by reversing the order of the elements in the original statement and negating both the hypothesis and conclusion. For example, the contrapositive of "If it rains, then the ground will be wet" is "If the ground is not wet, then it will not rain." How to use the converse, inverse ...
So now we know that the contrapositive, "If either m or n is odd, then m + n is odd," is false, because there is at least one case, 3 and 7, where the hypothesis is true but the conclusion is false. Remember that a statement like "<BLAH> is always true" can be proven false by just one example of when <BLAH> could be false.
2.2. Conditionals, Converse, Inverse, and Contrapositive. 🔗. Many statements and theorems in mathematics are of the form "If X is true, then Y is true.", called a conditional. 🔗. Conditionals are the basis for logical deduction, because we want to be able to make a conclusion based on known facts. 🔗.
A contrapositive is a statement that is formed by negating and swapping the hypothesis and the conclusion of an original conditional statement. It is a way of expressing the logical relationship between two statements. By understanding contrapositives, we can analyze the validity of arguments and identify equivalent statements.
Switching the hypothesis and conclusion of a conditional statement and negating both. For example, the contrapositive of "If it is raining then the grass is wet" is "If the grass is not wet then it is not raining." Note: As in the example, the contrapositive of any true proposition is also true. See also.
In this case, our hypothesis p is "a number is even," and our conclusion q is "a number is divisible by 2." Step 2: Identify the contrapositive of this statement.
A conditional statement (or 'if-then' statement) is a statement with a hypothesis followed by a conclusion. contrapositive: If a conditional statement is (if then ), then the contrapositive is (if not then not). converse: If a conditional statement is (if , then ), then the converse is (if , then . Note that the converse of a statement is not ...
Example. Conditional Statement: "If today is Wednesday, then yesterday was Tuesday.". Hypothesis: "If today is Wednesday" so our conclusion must follow "Then yesterday was Tuesday.". So the converse is found by rearranging the hypothesis and conclusion, as Math Planet accurately states. Converse: "If yesterday was Tuesday, then ...
This geometry video tutorial explains how to write the converse, inverse, and contrapositive of a conditional statement - if p, then q. This video also disc...