"Show that the empty set is a subset of every set." This is the problem I'm working on. It seems standard, but the nuance is that if two sets are disjoint, and the empty set is a subset of both, how do we draw the empty set in a Venn Diagram?
I searched and found this article: venn diagram of power set and empty set, but the person's response is that the empty set is in the intersection of all the other sets… my point is that those sets could be disjoint, so there would be no intersection. In that case, how do we draw it?
I recognize that the answer might just be that venn diagrams break down and don't illustrate the empty set. If so, are there other visualization techniques that are general enough? I know enough category theory to briefly think about it from that perspective.
Thanks in advance
Best Answer
You can draw a dot (circle with area zero) anywhere in the Venn Diagram and that can sort of serve as an intuition for the empty set. It contains nothing and consequently, has no area.
In fact, you can draw many dots spread out all over the Venn Diagram like you spilled glitter on your diagram. It doesn't matter how many you draw or where you draw them, as long as they are area zero.
You'll see that the empty set is, indeed, a subset of every set $A$, because the statement: "For every element $e$ in the empty set, $e \in A$" is vacuously true.