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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”

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