Step by step
The question
Question: prove that if n² + 3 is even, then n is odd.
1 Turn it around
An if-then statement means the same as its contrapositive: if not Q, then not P.
We saw it with truth tables.
Turned around, it says: if n is even, then n² + 3 is odd.
Starting from n is much easier: we know how to write it with letters.
Careful!
That swaps the two parts.
It is the converse, a different statement.
Turned around means: swap the parts and put not on both.
2 Write it with letters
n is even, so n = 2a, where a is a whole number.
That is what even means.
3 Put them in
Now put them in: n² + 3 becomes (2a)² + 3.
4 Work it out
(2a)² means the bracket times itself.
Every part times every part: 4a² + 3.
5 Show the form
We take out 2 from all but the 1: 2(2a² + 1) + 1.
2a² + 1 is a whole number, because whole numbers added and multiplied stay whole.
2 times a whole number, plus 1: that is the form of an odd number.
So n² + 3 is odd.
The turned statement is true, so the first one is true too: if n² + 3 is even, then n is odd.
The square at the end means: the proof is done.
6 Check
Try n = 1.
Then n² + 3 is 4: even.
And 1 is odd, as the statement says.
Right!
Let's look back.
First: turn it around.
Then: write it with letters.
Then: put them in.
Then: work it out.
Then: show the form.
Last: check.