Proof by Contrapositive help?

Question

I am having trouble providing a proof by contrapositive here, can anyone provide a proper proof by contrapositive here?

Let $x \in \mathbb{Z}$. Prove that $3x+1$ is even if and only if $5x-2$ is odd.

Answer 1

We shall prove that $3x + 1$ is even iff $5x - 2$ is odd by means of contraposition.

(Necessity, i.e., $5x - 2$ is odd implies $3x+1$ is even) Suppose $3x+1$ is odd, i.e., of the form $2j + 1$ for some integer $j$. Then $3x = 2j$. Now write

\begin{align*} 5x - 2 &= 2x + 3x - 2 \\ &= 2x + 2j - 2 \\ &= 2(x + j - 1) \end{align*}

and so $5x - 2$ is even, whence the desired result by contraposition.

(Sufficiency, i.e., $3x + 1$ even implies $5x - 2$ odd) Suppose $5x - 2$ is odd, i.e., of the form $2k + 1$ for some integer $k$. Now, employing a similar technique from above,

\begin{align*} 3x + 1 &= 5x - 2x + 1 - 3 \\ &= 5x - 2 - 2x \\ &= 2k + 1 - 2x \\ &= 2(k - x) + 1 \end{align*}

and so $3x + 1$ is odd, whence the desired result by contraposition. $\square$

Remark. Note that as we are discussing even and odd numbers, $x$ must necessarily be an integer.

Answer 2

Suppose $3x+1$ is even and $5x-2$ is also even, then $3x+1 = 5x-2 - [2(x-1)-1] = \text{even} - \text{odd}= \text{odd}$ , contradiction. On the other hand....assume $5x-2$ is odd and if $3x+1$ is odd, then $2x-3 = (5x-2) - (3x+1) = \text{even} - \text{even} = \text{even}$, contradiction to the fact that $2x-3 = 2(x-1)-1$ is clearly ..odd.

Answer 3

Use congruences mod. $2\,$: you want to prove that $$ x+1\equiv 0\mod 2\iff x\equiv 1 \mod 2, $$ and the contrapositive is \begin{align}&&x\not\equiv 1\mod 2 &\iff x+1\not\equiv 0\mod 2,\\ &\text{i.e. } \qquad &x\equiv 0 \mod 2&\iff x+1\equiv 1\mod 2, \end{align} which is obvious.

Answer 4

Proof: if $5x-2$ is even then $3x+1$ is odd

$5x-2$ is even $\implies 5x-2 = 2k$ for some $k \in \mathbb{Z}$

$\implies 5x=2(k+1) \implies 5x$ is even $\implies x$ is even

Then, $x = 2s$ for some $s \in \mathbb{Z}$

$\implies 3(2s) +1 = 6s+1 = 2(3s)+1$ is odd

Similarly, you can prove that if $3x+1$ is odd then $5x-2$ is even