Q:

Using a bijective function, prove there are equally many subsets of size $k$ as $n-k$

Let $S$ be a set with cardinality $n > 0$. Prove using a bijective function that the number of subsets of $S$ of size $k$ is equal to the number of subsets of $S$ of size $n−k$.

A:

For $A\subset S$, $\#A=k$, define $f(A)=S\setminus A$. 
This is a well-defined function, $f(A)$ is a set of size $\#S-\#A=n-k$, if $A_1\neq A_2$ ($k$-element subsets of $S$) then $f(A_1)\neq f(A_2)$.

