Year 11 Maths - Specialist Number and proof

Equivalent statements

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Questions

Question 1

56607

Prove by logical equivalence:

Let $n$ be a positive integer. $n-3$ is odd if and only if $n+2$ is even.

Answer

True

Worked Solution

$$\begin{aligned}
&\text{Let}\ n-3=2k+1\ \text{(k an integer)}\\
&\begin{aligned}
n &=2 k+4 \\
n+2 &=2 k+4+2 \\
&=2 k+6 \\
&=2(k+3)
\end{aligned}\\
&\therefore\ n+2\ \text{is even.}\end{aligned}$$

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