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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.

Thank you for your time.

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    $\begingroup$ Do you know how many elements $\emptyset$ has? $\endgroup$ – Michael Albanese Feb 6 '15 at 19:54
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    $\begingroup$ Hint: Is zero an odd or an even number? $\endgroup$ – Timbuc Feb 6 '15 at 19:55
  • $\begingroup$ 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? $\endgroup$ – Amy Feb 6 '15 at 20:04
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    $\begingroup$ 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. $\endgroup$ – Vladimir Vargas Feb 6 '15 at 20:08
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    $\begingroup$ math.stackexchange.com/questions/15556/is-zero-odd-or-even $\endgroup$ – Hans Lundmark Feb 6 '15 at 20:12
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Zero is an even number. It's $2 \times 0$.

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