There is question that I don't know how to prove. we have set $A=\{1,2,3,\ldots,n\},\; O=\{B\mid B⊆A,\text{ odd }B\},\; E=\{B\mid B⊆A,\text{ even }B\}$ it ask to prove that subsets even equal to subsets odd by proving that $f:O\to E$ is an injective and surjective function
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See also Exactly half of the elements of $\mathcal P(A)$ are odd-sized, Number of even and odd subsets, Number of subsets of even and odd – Martin Sleziak Jul 03 '15 at 12:53
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In general, one way to show that two sets have the same size is to construct a bijection between the two sets. So, find a function $f:E\to O$ which you can prove is a bijection. Hint: What happens if you fix some element in $A$ and add it to a given subset in $E$? What happens if you remove it from a given subset in $E$? If you answer these questions carefully you might get an idea of how to construct $f$. Then you can prove it is a bijection.
Ittay Weiss
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Let $$ f(B) = \begin{cases} B\smallsetminus\{1\} & \text{if }1\in B, \\ B\cup\{1\} & \text{if }1\not\in B. \end{cases} $$ This works unless $n=0$.
