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If we have a function $f:A \rightarrow B$, then one way to give meaning, I think, to this function, in terms of set theory, is to say, that $f$ is actually a binary relation $f=(A,B,G_f)$, where $G_f \subseteq A \times B$ is the graph of the function. Now my question is: what is $f$ if

$\bullet \ A=\emptyset, \ B\neq\emptyset$,?

$ \bullet \ B=\emptyset, \ A\neq\emptyset$ ?

$ \bullet \ B=\emptyset, \ A=\emptyset$ ?

(Another way to formulate this, I think, would be: How do the sets $\emptyset\times B,\ A\ \times \emptyset, \ \emptyset \times \emptyset $ look like ? Are they all $\emptyset$ ?)

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2 Answers 2

up vote 14 down vote accepted

Yes, they're all empty sets. For example, $\emptyset \times A$ consists of all pairs of the form $(o,a)$ with $o \in \emptyset, a \in A$. But the empty set has no elements, hence $\emptyset \times A$ has no elements, hence $\emptyset \times A$ is the empty set. A similar argument works for the other two sets.

Here is how this problem can be interpreted in terms of cardinalities. For any sets $A,B$ the cardinality of $A \times B$ is the product of cardinalities of $A$ and $B$. Hence the cardinality of $\emptyset \times A$ is just $0 \cdot |A| = 0$ so $\emptyset \times A$ has $0$ elements, and hence $\emptyset \times A = \emptyset$. And a similar argument will work in the other two cases.

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Although I understand well what you mean, isn't writing $o\in \emptyset$ a bit weird, since the emptyset should not contain any other set ? –  temo Jul 14 '11 at 9:07
Yes, you are correct. I just used the definition of Cartesian product to write down how an element of $\emptyset \times A$ "looks like". And since $\emptyset$ has no elements, the condition $o \in \emptyset$ is never satisfied, hence $\emptyset \times A$ is empty. Basically, I was trying to say "the empty set has no elements, hence $\emptyset \times A$ has no elements" rigorously. –  algebra_fan Jul 14 '11 at 9:15
The first answer is spot on, but I don't see a way of formalizing the second that doesn't pass through the first, especially if $B$ is infinite (but even if $B$ is finite, there's something to show). –  user83827 Jul 14 '11 at 10:01
Well, I am using the arithmetic of cardinal numbers; see en.wikipedia.org/wiki/Cardinal_number#Cardinal_multiplication. So if you are familiar with this, then you can just use the definition of cardinal multiplication in this problem. But I agree that it's certainly overkill to use this machinery of cardinal numbers to answer this question. –  algebra_fan Jul 14 '11 at 10:18
It's not that I think it's overkill. I think it's begging the question argue that $\emptyset \times B$ is empty since $0 \times \kappa = 0$ for all $\kappa$. Like I said, I think the first answer is spot on, and I'm sorry if this seems pedantic, but I think it's misleading to suggest that changing the language of the question to an equivalent statement in cardinal arithmetic (and implicitly appealing to an unjustified "$0$ times anything is $0$" intuition) is a suitable replacement for the actual argument. –  user83827 Jul 14 '11 at 10:37

A slightly more formal rephrase of the other answer is to calculate which elements are in $A \times \emptyset$: for any $p$, \begin{align} & p \in A \times \emptyset \\ \equiv & \;\;\;\;\;\text{"definition of $\times$ on sets"} \\ & \textrm{isPair}(p) \;\land\; \textrm{fst}(p) \in A \;\land\; \textrm{snd}(p) \in \emptyset \\ \equiv & \;\;\;\;\;\text{"$x \in \emptyset \equiv \textrm{false}$ for any $x$; simplify"} \\ & \textrm{false} \\ \equiv & \;\;\;\;\;\text{"$x \in \emptyset \equiv \textrm{false}$ for any $x$"} \\ & p \in \emptyset \\ \end{align} and therefore, by set extensionality, $A \times \emptyset = \emptyset$. A very similar proof of course goes for $\emptyset \times B = \emptyset$, and obviously either of these implies $\emptyset \times \emptyset = \emptyset$.

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