How many subsets with an even number of elements?

Question

I don't quite understand this topic so I need help to prove the number of subsets of

S = {1,2,3,...,N}

which they have an even number of elements.

Thanks in advance.

Why not work it out for small $N$? I suspect the pattern will emerge. — lulu, Apr 20 '20 at 22:36

score 0 · Accepted Answer · 2020-04-20T23:06:33.700

0

There are $2^N$ subsets altogether. Half should have an even number of elements. That's $1/2\cdot2^N=2^{N-1}$.

You could do a proof by induction.

edited Apr 20 '20 at 23:06

answered Apr 20 '20 at 22:40

"Half should have an even number of elements." Why? You shouldn't claim half of things are even unless you have a reason to believe even and odd number of items occur equally often. (How many even primes are there? Primes can either be even or odd so it makes sense that half of them are even so that means there are infinitely many even primes, doesn't it?) – fleablood Apr 20 '20 at 23:28

1 Answers1