Constructing Proofs: Direct & Indirect Methods

Constructing Proofs with Direct and Indirect Methods

In mathematics, constructing proofs is a fundamental skill that validates the truth of statements based on logical reasoning. Two primary methods of constructing proofs are direct and indirect methods. Direct proofs proceed by straightforward reasoning to establish the truth of a statement, while indirect proofs, such as proof by contradiction, assume the negation of a statement to demonstrate its falsehood. Below, we present three diverse and practical examples to illustrate these methods.

Example 1: Proving the Sum of Two Even Numbers is Even

Context

This example demonstrates a direct proof related to number theory, focusing on properties of even numbers.

When we define even numbers, we know that any even number can be expressed as 2n, where n is an integer. This property will help us prove that the sum of two even numbers is also an even number.

The proof is as follows: Let’s take two even numbers, say a and b. By definition of even numbers, we can write:

• a = 2m (where m is an integer)
• b = 2n (where n is also an integer)

Now, we add these two numbers together:

• a + b = 2m + 2n
• = 2(m + n)

Since m and n are integers, (m + n) is also an integer. Thus, we can express the sum as 2k, where k = m + n. Therefore, a + b is even.

Notes

• Variations of this proof can be made by changing the integers involved, or by extending the concept to three or more even numbers.
• This serves as a foundational property in algebra and is often used in proofs concerning divisibility.

Example 2: Proving the Contrapositive of a Statement

Context

In this example, we will use an indirect method to prove a statement involving implications. The contrapositive of an implication is logically equivalent to the original statement.

Let’s prove the statement: If a number is odd, then its square is odd. The contrapositive of this statement is: If the square of a number is even, then the number itself is even. We will prove this contrapositive.

Assume that the square of a number, say x, is even. Thus, we can express this as:

• x² = 2k (for some integer k)

Taking the square root gives:

• x = √(2k)

Now, since x² is even, it follows that x must also be even, which can be represented as:

• x = 2m (for some integer m)

Thus, we conclude that if the square of a number is even, then that number must be even. This verifies the contrapositive and, by extension, confirms that if a number is odd, then its square is odd.

Notes

• A good understanding of logical equivalences is crucial for applying proof by contraposition effectively.
• This method is often used in advanced proofs and can simplify complex logical reasoning.

Example 3: Proof by Contradiction for the Irrationality of √2

Context

This example illustrates an indirect method known as proof by contradiction, often used in real analysis and number theory. We aim to prove that √2 is irrational.

Assume the opposite: that √2 is rational. This means we can express it as a fraction in its simplest form:

• √2 = a/b (where a and b are integers with no common factors, and b ≠ 0)

Squaring both sides gives:

• 2 = a²/b²
• Thus, a² = 2b²

This implies that a² is even, which means a must also be even (since the square of an odd number is odd). If a is even, we can express it as:

• a = 2k (for some integer k)

Substituting back, we have:

• (2k)² = 2b²
• 4k² = 2b²
• This simplifies to b² = 2k²

Hence, b² is also even, which implies b is even as well. Therefore, both a and b are even, contradicting our original assumption that they had no common factors. Thus, our assumption that √2 is rational must be false; therefore, √2 is irrational.

Notes

• Proof by contradiction is a powerful technique and can be applied to a wide range of mathematical statements.
• Understanding the properties of rational and irrational numbers is key to using this method effectively.