Example of a Direct Proof
Let's consider the following proposition:
"For all positive integers n, if n is even, then n2 is even."
To prove this proposition using a direct proof, we can follow these steps:
1. Clearly state the proposition: "For all positive integers n, if n is even, then n2 is even."
2. Assume the premise: Let's assume that n is a positive even integer.
3. Use the definition of even numbers: An even number can be expressed as 2k, where k is an integer.
4. Substitute n with 2k in the proposition: We have n = 2k, and we need to prove that (2k)2 is even.
5. Simplify the expression: (2k)2 = 4k2, which is 2(2k2).
6. By definition, 2(2k2) is an even number since it can be expressed as 2m, where m = 2k2.
7. Thus, we have shown that if n is even (represented as n = 2k), then n2 (represented as (2k)2) is even.
8. Conclude the proof: We have successfully proven the proposition that for all positive integers n, if n is even, then n2 is even.
In this example, we used the definition of even numbers and algebraic manipulations to establish the truth of the proposition through a direct proof.