Proof By Cases---Week 6

Sometime, it is more convenience to prove some proofs by cases. Consider the following proof:

"for all n belongs to N, n^2+n is even"

in order to prove above, it is more convenience to consider the case where n is odd ,then the case n is even.

First, we consider n is odd, then we can use definition of odd to assume n is of the form n=2k+1,where k is natural number. Then n^2+n=(2k+1)(2k+2)=2(k+1)(2k+1)=2m ,where m is a natural number of the form m=(k+1)(2k+1), therefore by definition  n^2+n is also even.

second, we consider n is even, then we can again use definition of even to assume n is of the form n=2k, where k is natural number. Then  n^2+n=(2k)(2k+1)=2n ,where n is a natural number of the form n=k(2k+1), so by definition n^2+n is also even.

So, no matter n is odd or even, we conclude that n^2+n is  even.

Finally, proof by cases are  good techniques that can make some proofs become so easy , therefore we need to know when or how to use proof by cases in the future.