1. ## Implication theorem

The following theorem has the form of an implication.

If n is an integer and 3n+2 is even, then n is even.

a) Prove this theorem by contradiction.
b) Give a direct proof of this theorem.
(Hint: it may be useful to get an equation for n in which
the term to which n is equal also includes n itself.)

2. Originally Posted by kashifzaidi
The following theorem has the form of an implication.

If n is an integer and 3n+2 is even, then n is even.

a) Prove this theorem by contradiction.
b) Give a direct proof of this theorem.
(Hint: it may be useful to get an equation for n in which
the term to which n is equal also includes n itself.)

Let $\displaystyle n\in\mathbb{Z}$ and let $\displaystyle 3n+2$ be even.

Assume $\displaystyle n$ is not even. Therefore $\displaystyle n$ is odd. Therefore $\displaystyle n=2x+1$ for some $\displaystyle x\in\mathbb{Z}$

So $\displaystyle 3n+2=3(2x+1)+2=6x+3+2=6x+5=2(3x+2)+1$

But $\displaystyle x$ is an integer, so $\displaystyle 3x$ is an integer, so $\displaystyle 3x+2$ is an integer. Thefore $\displaystyle 2(3x+2)+1=2z+1$ for some $\displaystyle z\in\mathbb{Z}$

This is the definition of an odd number, so $\displaystyle 3n+2$ is odd. But we know $\displaystyle 3n+2$ is even. This is a contradiction and therefore the assumption that $\displaystyle n$ is not even is wrong. Therefore $\displaystyle n$ is even

Can you give the second part b try