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# Demonstration Techniques: The Contrapositive

The contraposition is one of the most used demonstration techniques. Let's see together its principles and some common applications where it is used.

## What is the proof by contraposition?

The proof by contraposition is based on the following principle: The proposition “A implies B” is equivalent to the proposition “Not B implies not A”. Mathematically this is written:

( A \Rightarrow B) \iff ( \lnot B \Rightarrow \lnot A)

Why this principle? Because sometimes it is easier to show “not B implies not A” than to show directly “A implies B”.

The first example of a contrapositive that we learn in mathematics is the Pythagorean theorem. Indeed, the statement of the Pythagorean theorem is as follows: "if a triangle is right-angled, then the square of the length of its hypotenuse - the side opposite its right angle - is equal to the sum of the squares of the lengths of the two sides forming the right angle.

Its contrapositive is then: "If the square of the longest side of a triangle is not equal to the sum of the squares of the two other sides, then this triangle is not right-angled"

In everyday life, an example is the contrapositive of the following proposition "If it is raining then I have an umbrella" whose contrapositive "is" If I do not have an umbrella then it is not raining.

## Examples of proof by contraposition

### Example 1

We consider an integer n. We want to show the following proposition: “If n2 is odd, then n is odd. For this, we will show the contrapositive: “If n is even, then n2 is even”.

Proof: There exists k integer such that n = 2k. Squaring, we get:

n^2 = (2k)^2 = 4k^2 = 2 \times(2k^2)

So n2 is even. It has been proven that “If n is even, then n2 is even” and therefore that “If n2 is odd, then n is odd.

### Example 2

To show the following property:
Yes ton – 1 is prime so a = 2 and n is prime.
To do this, we then show the contrapositive
¬B is written “a ≠ 2 or n is not prime”
¬A is written “an – 1 is not prime”

If you are interested in the demo,