# Can the empty set be considered to have an even or odd number of elements?

Here's the problem:

Let $A = \{1,2,3,4,5\}$ and define a function $F: \mathcal{P}(A) \to \mathbb{Z}$ as follows: For all sets $X \in \mathcal{P}(A)$,

$$F(x) = \begin{cases} 0 &\ \text{if X has an even number of elements}\\ 1 &\ \text{if X has an odd number of elements}. \end{cases}$$

Find $F(\emptyset)$.

So I'm hoping someone can help me determine whether nothing is itself an element, and, if not, whether the absence of an element is even or odd.

• Do you know how many elements $\emptyset$ has? – Michael Albanese Feb 6 '15 at 19:54
• Hint: Is zero an odd or an even number? – Timbuc Feb 6 '15 at 19:55
• There are no elements in an empty set, by definition. However, I don't know whether nothing is even or odd... can I say that nothing and 0 are the same thing? – Amy Feb 6 '15 at 20:04
• Amy, there's nothing in the empty set, which means that there are $0$ elements. If $0$ were to be an element, then the set would have $1$ element. – Vladimir Vargas Feb 6 '15 at 20:08
• math.stackexchange.com/questions/15556/is-zero-odd-or-even – Hans Lundmark Feb 6 '15 at 20:12

Zero is an even number. It's $2 \times 0$.