1

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.

Amy
  • 13
  • 1
    Do you know how many elements $\emptyset$ has? – Michael Albanese Feb 06 '15 at 19:54
  • 3
    Hint: Is zero an odd or an even number? – Timbuc Feb 06 '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 06 '15 at 20:04
  • 1
    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 06 '15 at 20:08
  • @Amy: 'Nothing' is not a number so it does not make sense to say whether it is even or odd. I would advise against using the word to describe the empty set. As you have established, the empty set has zero elements, so to determine $F(\emptyset)$ you just need to determine whether $0$ is even or odd. – Michael Albanese Feb 06 '15 at 20:08
  • 1
    http://math.stackexchange.com/questions/15556/is-zero-odd-or-even – Hans Lundmark Feb 06 '15 at 20:12

1 Answers1

0

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

TonyK
  • 64,559