Let’s refresh the concept of direct proof.
“A direct proof of a conditional statement p → q is constructed when the first step is the assumption that p is true; subsequent steps are constructed using rules of inference, with the final step showing that q must also be true. A direct proof shows that a conditional statement p → q is true by showing that if p is true, then q must also be true, so that the combination p true and q false never occurs.” Discrete Mathematics and its Applications by Rosen.
Let n be an even number.
n=2k for some integer k.
The additive inverse of n is -n.
If n=2k, then
Therefore, -n is also an even number.
- 3. Show that the square of an even number is an even number using a direct proof
- 5. Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind of proof did you use?
- 6- Use a direct proof to show that the product of two odd numbers is odd
- 7. Use a direct proof to show that every odd integer is the difference of two squares.