Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contraposition. b) a proof by contradiction.

Solution:

We have to prove that if n is an integer and 3n + 2 is even, then n is even using

a)  Proof by contraposition

A proof by contrapositive means that we will prove the opposite of the given statement.

In this case, we have to prove that when n is odd, then 3n + 2 is odd

Assume n is odd, n = 2m + 1

Where, m is an integer.

The assumed n in the given expression

3n + 2 = 3(2m+1) + 2

= 6m + 3 + 2

= 6m + 5

Rewriting the expression,

= (6m + 4) + 1

= 2(3m + 2) + 1

Now, 2(3m + 2) implies that multiplying a number (3m + 2) by 2 will always give an even number.

Adding one to the result means the final result is odd.

When n is odd, the expression 3n + 2 is odd.

b) Proof by contradiction

A proof by contradiction means we will prove that if the given statement is F, then this will lead to a contradiction.

For the given statement 3n + 2 to be F, 3n + 2 has to be even while n is odd.

Assume 3n + 2 is even and n is odd.

Substitute with the assumed n in the given expression,

3n + 2 = 3(2m + 1) + 2

= 6m + 3 + 2

= 6m + 5

Rewriting the expression,

= (6m + 4) + 1

= 2(3m + 2) + 1

Now, 2(3m + 2) implies that multiplying a number (3m + 2) by 2 will always give an even number.

Adding one to the result means the final result is odd.

The expression implies that 3m + 2 is odd which contradicts the assumption 3n + 2 is even.

Therefore,

a) proof by contraposition implies that when n is odd, the expression 3n + 2 is odd.

b) proof by contradiction implies that 3m + 2 is odd which contradicts the assumption 3n + 2 is even.

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Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contraposition. b) a proof by contradiction.

Summary:

If n is an integer and 3n + 2 is even, then n is even using (a) a proof by contraposition implies that when n is odd, the expression 3n + 2 is odd. (b) a proof by contradiction implies that 3m + 2 is odd which contradicts the assumption 3n + 2 is even.