Step by step
The question
Question: prove that if x − y is odd, then x is odd or y 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 x and y are both even, then x − y is even.
Starting from x and y is much easier: we know how to write them 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
x is even, so x = 2a, where a is a whole number.
That is what even means.
y is even, so y = 2b, where b is a whole number.
A new letter, b: y can be a different number from x.
3 Put them in
Now put them in: x − y becomes (2a) − (2b).
4 Work it out
Open the brackets: 2a − 2b.
5 Show the form
Every term has a 2 in it.
We take out 2: 2(a − b).
a − b is a whole number, because whole numbers added and multiplied stay whole.
2 times a whole number: so x − y is even.
The turned statement is true, so the first one is true too: if x − y is odd, then x is odd or y is odd.
The square at the end means: the proof is done.
6 Check
Try x = 3 and y = 5: it gives -2.
-2 is not odd, so the statement says nothing here.
Fine.
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.