3 Practical Examples of Direct Proof

Understanding Direct Proof

Direct proof is a fundamental technique in mathematics used to establish the truth of a statement. It involves a straightforward chain of logical deductions from known facts or axioms to reach a conclusion. This method is particularly effective in proving theorems, as it allows one to build a clear and methodical argument. Below are three diverse examples demonstrating the application of direct proof.

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

Context

This example illustrates how to prove a basic property of even numbers, which is often used in number theory and algebra.

To prove that the sum of two even integers is even, we start by defining what it means for a number to be even.

Let’s denote two even integers as 2a and 2b, where a and b are integers.

By definition, an integer is even if it can be expressed as 2 times another integer. Therefore:

• The first even integer can be expressed as: 2a
• The second even integer can be expressed as: 2b

Now, we will add these two integers:

• The sum is 2a + 2b
• This can be factored as: 2(a + b)

Since a + b is an integer (the sum of two integers is always an integer), we see that the result, 2(a + b), is also an even integer. Hence, we have proved that the sum of two even integers is even.

Notes

• This proof can be extended to show that the sum of any even number of even integers is even.
• A similar proof can be constructed for odd integers, leading to the conclusion that the sum of two odd integers is also even.

Example 2: Proving that the Square of an Odd Integer is Odd

Context

In this example, we will prove a property of odd integers, which is useful in various mathematical problems and proofs.

Let’s denote an odd integer as 2k + 1, where k is an integer. We need to prove that the square of this integer is also odd.

To find the square, we perform the following calculation:

• The odd integer can be expressed as: (2k + 1)
• Now, we calculate its square: (2k + 1)²

Expanding this gives:

• (2k + 1)² = 4k² + 4k + 1
• This can be factored as: 2(2k² + 2k) + 1

Here, 2k² + 2k is an integer (since k is an integer, both 2k² and 2k are integers, and their sum is also an integer). Therefore, we can conclude that 2(2k² + 2k) + 1 is of the form 2m + 1, where m is an integer. Thus, we have shown that the square of an odd integer is indeed odd.

Notes

• This property of odd integers is often leveraged in proofs involving parity.
• It can also be used to prove that the product of two odd integers is odd as well.

Example 3: Proving that the Product of Two Positive Integers is Positive

Context

This example demonstrates a fundamental property of positive integers, which is applicable in various mathematical fields including algebra and calculus.

Let’s denote two positive integers as x and y. By definition, positive integers are greater than zero. We want to prove that the product xy is also positive.

Since x and y are positive, we have:

• x > 0
• y > 0

Now, we can multiply these inequalities:

• xy > 0

This follows from the property that the product of two positive numbers is always positive. Hence, we have successfully shown that the product of two positive integers is positive.

Notes

• This proof can be extended to show that the product of any number of positive integers is positive.
• A related concept is that if either integer is zero or negative, the product will not remain positive, leading to further explorations in number theory.

In summary, these examples of direct proof highlight how clear logical deductions can be used to establish mathematical truths effectively.