Question

Use a direct proof to show this. You may have to divide your proof into cases.

a. If n∈Z, then n²+3n+4 is even.
b. Suppose a,b∈Z. If a²(b²−2b) is odd, then both a and b are odd.
c. If a is odd or b is even, then (a−3)b² is even.

Answer 1

To prove these statements, we use direct proofs. In a direct proof, we assume the premise is true and show that the conclusion must also be true.

Step-by-step explanation:

To prove a statement using a direct proof, we assume that the premise is true and then show that the conclusion must also be true. Let's address each part of the question:

a. If n∈Z, then n²+3n+4 is even:

We need to prove that n²+3n+4 is divisible by 2. We can rewrite the expression as n(n+3)+4. Since n and (n+3) are consecutive integers, one of them must be even. Therefore, their product is even. Adding 4 does not change the even nature of the expression, so n²+3n+4 is even.

b. Suppose a,b∈Z. If a²(b²−2b) is odd, then both a and b are odd:

We can rewrite the expression as a²(b(b−2)). We know that the product of two integers is odd only if both integers are odd. Therefore, for a²(b(b−2)) to be odd, both a and b must be odd.

c. If a is odd or b is even, then (a−3)b² is even:

If a is odd, then (a−3) is even. If b is even, then b² is even. Therefore, the product of an even number and an even number is always even.