In order to do mathematics, we need to understand some basic proof techniques. The two simplest techniques are:
- In order to show that an existential statement is true: give an example
- In order to show that a universal statement is false: give a counter-example
When we try to find a proof based on logical deductions, the above techniques may be a bit complicated. in such cases, a technique called proof by contradiction can be useful. In order to show that a statement is true, assume the opposite and show that it leads to a false conclusion.
Let’s solve some sums based on proof by contradiction.
- Use the method of proof by contradiction to show that 200 is not a perfect square.
Let us assume that 200 is a perfect square. So, n*n=200 where n is a positive integer.
We know that, 14*14=196 and 15*15=225. So, based on this we can deduce that 14<n<15. So, n is not an integer.
So, we conclude that 200 is not a perfect square.
2. Prove that, for all natural numbers n, n*n+7n+12 is an even number.
Let us assume that for all natural numbers n, n*n+7n+12 is not an even number. So, it is an odd number.
If we take n=1, result is 20 i.e. even number. Similarly for n=2, result is 30 i.e. even number and so on. So, it is not an odd number.
So, we conclude that n*n+7n+12 is an even number.
3.State whether for all natural numbers n, 7n+2 is a perfect square is true or false.
For n=1, result is 9 i.e. a perfect square. Similarly for n=2, result is 16 i.e. a perfect square. When n=3, the result is 23 i.e. not a perfect square. So, the statement is false.
So, we conclude that n=3 is the counter example for 7n+2.