Worked examples
The question
Question: prove that if n² is even, then n is even.
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.
if P, then Q ≡ if not Q, then not P
Turned around, it says: if n is odd, then n² is odd.
Starting from n is much easier: we know how to write it with letters.
turned around: if n is odd, then n² is odd
Careful!
That swaps the two parts.
It is the converse, a different statement.
Turned around means: swap the parts and put not on both.
Careful: not if n is even, then n² is even ✗
2 Write it with letters
n is odd, so n = 2a + 1, where a is a whole number.
n = 2a + 1
That is what odd means.
odd = 2 × a whole number + 1
3 Put them in
Now put them in: n² becomes (2a + 1)².
n2 = (2a + 1)2
4 Work it out
(2a + 1)² means the bracket times itself.
= (2a + 1)(2a + 1)
Every part times every part: 4a² + 2a + 2a + 1.
= 4a2 + 2a + 2a + 1
Like terms together: 4a² + 4a + 1.
= 4a2 + 4a + 1
5 Show the form
We take out 2 from all but the 1: 2(2a² + 2a) + 1.
= 2(2a2 + 2a) + 1
2a² + 2a is a whole number, because whole numbers added and multiplied stay whole.
2a² + 2a is a whole number
2 times a whole number, plus 1: that is the form of an odd number.
So n² is odd.
So n² is odd.
The turned statement is true, so the first one is true too: if n² is even, then n is even.
The square at the end means: the proof is done.
So: if n² is even, then n is even. ∎
6 Check
Try n = 2.
Then n² is 4: even.
n = 2: n² = 4, even
And 2 is even, as the statement says.
Right!
and 2 is even
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.
Answer 2a² + 2a is a whole number
▶ Watch it solved
The question
Question: prove that if 3n + 2 is odd, 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.
if P, then Q ≡ if not Q, then not P
Turned around, it says: if n is even, then 3n + 2 is even.
Starting from n is much easier: we know how to write it with letters.
turned around: if n is even, then 3n + 2 is even
Careful!
That swaps the two parts.
It is the converse, a different statement.
Turned around means: swap the parts and put not on both.
Careful: not if n is odd, then 3n + 2 is odd ✗
2 Write it with letters
n is even, so n = 2a, where a is a whole number.
n = 2a
That is what even means.
even = 2 × a whole number
3 Put them in
Now put them in: 3n + 2 becomes 3(2a) + 2.
3n + 2 = 3(2a) + 2
4 Work it out
Open the brackets: 6a + 2.
= 6a + 2
5 Show the form
Every term has a 2 in it.
We take out 2: 2(3a + 1).
= 2(3a + 1)
3a + 1 is a whole number, because whole numbers added and multiplied stay whole.
3a + 1 is a whole number
2 times a whole number: so 3n + 2 is even.
So 3n + 2 is even.
The turned statement is true, so the first one is true too: if 3n + 2 is odd, then n is odd.
The square at the end means: the proof is done.
So: if 3n + 2 is odd, then n is odd. ∎
6 Check
Try n = 1.
Then 3n + 2 is 5: odd.
n = 1: 3n + 2 = 5, odd
And 1 is odd, as the statement says.
Right!
and 1 is odd
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.
Answer 3a + 1 is a whole number
▶ Watch it solved
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.
if P, then Q ≡ if not Q, then not P
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.
turned around: if x and y are both even, then x + y is even
Careful!
That swaps the two parts.
It is the converse, a different statement.
Turned around means: swap the parts and put not on both.
Careful: not if x is odd or y is odd, then x + y is odd ✗
2 Write it with letters
x is even, so x = 2a, where a is a whole number.
x = 2a
That is what even means.
even = 2 × a whole number
y is even, so y = 2b, where b is a whole number.
A new letter, b: y can be a different number from x.
y = 2b
3 Put them in
Now put them in: x + y becomes (2a) + (2b).
x + y = (2a) + (2b)
4 Work it out
Open the brackets: 2a + 2b.
= 2a + 2b
5 Show the form
Every term has a 2 in it.
We take out 2: 2(a + b).
= 2(a + b)
a + b is a whole number, because whole numbers added and multiplied stay whole.
a + b is a whole number
2 times a whole number: so x + y is even.
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.
So: if x + y is odd, then x is odd or y is odd. ∎
6 Check
Try x = 3 and y = 5: it gives 8.
x = 3, y = 5: x + y = 8, even
8 is not odd, so the statement says nothing here.
Fine.
Right!
and 3 is odd
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.
Answer a + b is a whole number
▶ Watch it solved
