Introduction to Proof by Contrapositive

Proof by contrapositive is a powerful technique in mathematical reasoning. It involves proving that if a statement implies another, then the negation of the second statement implies the negation of the first. This method is particularly useful when the direct proof of a statement is complex or convoluted. Instead of proving the statement directly, we can prove its contrapositive, which is logically equivalent. Below are three diverse examples that illustrate this proof strategy.

Example 1: Even Numbers and Their Squares

Context

This example demonstrates how to prove a property about even numbers and their squares using proof by contrapositive.

If we want to show that if a number is even, then its square is even, we can instead prove the contrapositive: if the square of a number is not even, then the number itself is not even.

We start with the statement: If n is an even number, then n² is even.

The contrapositive would be: If n² is odd, then n is odd.

To prove this, we assume that n² is odd. From the definition of odd numbers, we can express n² as n² = 2k + 1 for some integer k. Now we can consider the implications:

• Since n² = 2k + 1, it cannot be expressed in the form of 2m, which represents even numbers.
• The only way for n² to be odd is if n itself must also be odd, as the square of an even number will always be even (i.e., n = 2m).

Thus, we conclude that if n² is odd, then n must also be odd, proving the contrapositive and, consequently, the original statement.

Notes

This example highlights the relationship between odd and even numbers, reinforcing the understanding of number properties.

Example 2: Implications of Triangles

Context

In geometry, we often deal with properties of shapes. Here, we will prove a statement regarding triangles using proof by contrapositive.

We want to show: If a triangle is equilateral, then it has three equal angles. Instead, we will prove the contrapositive: If a triangle does not have three equal angles, then it is not equilateral.

Assuming that a triangle does not have three equal angles means at least one angle is different from the others. If we have a triangle with angles A, B, and C, we denote one angle, say A, as differing from the others, leading us to:

• The angles of a triangle sum up to 180 degrees, so if A is different, it creates a scenario where the triangle cannot be equilateral.
• An equilateral triangle requires all angles to be equal (60 degrees each), which contradicts our assumption of having at least one unequal angle.

Thus, we conclude that if a triangle does not have three equal angles, it cannot be equilateral, thereby proving the contrapositive and the original statement.

Notes

This example is beneficial for students learning about triangle properties and helps reinforce foundational concepts in geometry.

Example 3: Divisibility and Prime Numbers

Context

This example illustrates the application of proof by contrapositive in relation to prime numbers and their properties.

We aim to show: If p is a prime number, then p has no divisors other than 1 and p itself. We will instead prove the contrapositive: If p has a divisor other than 1 and p, then p is not a prime number.

Assuming p has a divisor d such that 1 < d < p, it follows that:

• By the definition of a prime number, it can only be divided evenly by 1 and itself. Therefore, the existence of d means p can be divided by a number other than 1 and p, violating the prime definition.
• Consequently, if p has a divisor other than 1 and itself, it cannot be classified as a prime number.

Thus, we conclude that if p has a divisor other than 1 and p, then it is not prime, proving the contrapositive and thus the original statement.

Notes

This example serves as a fundamental concept in number theory and is particularly useful for students studying prime numbers and divisibility rules.