Subconjuntos e hiperconjuntos

So what does it mean for a set to be a subset of another set?

Essentially a subset is any set that is completely contained in the other set.

Right now you're probably thinking, "Okay, you reworded it. But you still haven't told me what containment means!"

And I'm getting there, but thinking about it in terms of containment is more intuitive.

Before I actually define containment, let me just ask you to think about it a bit yourself.

Now that you hopefully have a bit of a feel for what we're going for, you should understand the definition better.

A set, $X$, is a subset of (or is contained in) a set, $Y$, if every element of $X$ is also an element of $Y$. This is written $X \subset Y$.

Therefore $\{1,2,3\}\subset \{1,2,3,4\}\subset \mathbb{Z}$ since $1,2,$ and $3$ are all in $\{1,2,3,4\}$, and $1,2,3,$ and $4$ are all integers.

So now we know what a subset is, so what's a superset? Well, it's just the opposite: if $A\subset B$, then B is a superset of A, written $B\supset A$

Note that the empty set is a subset of every set, since all of it's elements are always in every set, since it has no elements. (Yeah I know. It's a bit dumb, but it's for convenience just like the empty set itself.)

Also every set is a subset of itself, since a set by definition contains all the elements it contains.

What are all the subsets of the set $\{x,y,z\}$? Take a moment to work it out.

These are the subsets: $$\mathcal{P}\left(\{x,y,z\}\right)=\left\{\varnothing,\{x\},\{y\},\{z\},\{x,y\},\{y,z\},\{x,z\},\{x,y,z\},\right\}$$

Whoa wait!! What's that $\mathcal{P}$ thing doing?

It turns out that $\mathcal{P}$ stands for the Power set of a set. The power set of a set is the set of all subsets of that set, as demonstrated above.