contrapositive hypothesis conclusion

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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."

If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

In the above example, since the hypothesis and conclusion are equivalent, all four statements are true. But this will not always be the case!

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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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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

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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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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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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