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A bit of homework that I'm not sure on. The question reads:

Let $A=\{a\}$ and $B=\{1,2\}$. Find the following:

$$\mathcal{P}(A) \times \mathcal{P}(B)$$

The worked out solution is as follows.

$\{ (\emptyset, \emptyset), (\emptyset, \{1\}), (\emptyset, \{2\}), (\emptyset, \{1, 2\}), (\{a\}, \emptyset\}), (\{a\}, \{1\}), (\{a\}, \{2\}), (\{a\}, \{1, 2\}) \}$

I know that $(\emptyset, \{1\})$ would make a horizontal line at the $y$-coordinate of $\{1\}$ (right?), but I'm not sure what would be drawn on a plane at $(\emptyset, \emptyset)$.

Any help would be appreciated. Thanks!

Michael Hardy
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Tim
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    These are not points in a plane, because the elements of each pair are sets, not numbers. Does the question tell you to plot it on some plane? You could make a plane with subsets of $A$ along one axis and those of $B$ along the other. This would be a "discrete" plane. And $(\phi, {1})$ will be a point, not a line. – M. Vinay Jun 10 '14 at 03:51

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It is not necessary (and in my opinion, not very helpful) to consider the ordered pairs as subsets of the two-dimensional space $\mathbb{R}^2$. I think it is better to think of the elements of $A$ and $B$ as just arbitrary objects.

That aside, the pair $(\varnothing, \{1\})$ would not be a horizontal line; that would be $(\mathbb{R},\{1\})$. The set $(\varnothing, \{1\})$ would not appear on the two-dimensional space $\mathbb{R}^2$, that is, it is unclear how to represent it on the plane. Likewise, it is unclear how to represent $(\varnothing,\varnothing)$ on the plane. This is why I suggested that you think of the elements of $A$ and $B$ not as real numbers, but as objects (apples, oranges, etc.), and the ordered pairs as simply that: pairs.

angryavian
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