Number the members of the set $1,2,3,4,\ldots,n$.
For every subset with an even number of elements, there is a corresponding set with an odd number of elements, that corresponds in this way:
- If $1$ is a member of the set with an even number of elements, then delete $1$ from the set to get a set with an odd number of elements.
- If $1$ is not a member of the set with an even number of elements, then add $1$ to the set to get a set with an odd number of elements.
For example, suppose the set is $\{1,2,3,4\}$. Then we have this correspondence between sets with an even number of elements and sets with an odd number of elements:
$$
\begin{array}{rcl}
\text{even} & & \text{odd} \\
\hline
\varnothing & \leftrightarrow & \{1\} \\
\{1,2\} & \leftrightarrow & \{2\} \\
\{1,3\} & \leftrightarrow & \{3\} \\
\{1,4\} & \leftrightarrow & \{4\} \\
\{2,3\} & \leftrightarrow & \{1,2,3\} \\
\{2,4\} & \leftrightarrow & \{1,2,4\} \\
\{3,4\} & \leftrightarrow & \{1,3,4\} \\
\{1,2,3,4\} & \leftrightarrow & \{2,3,4\}
\end{array}
$$
This won't work with the empty set because we don't have an element to which we can assign the number $1$.