Cardinality of a Set

The cardinality of a set is the number of elements in the set. It gives us a way to describe whether two sets are the same size. For a set $S$, we write the cardinality as $|S|$.

For finite sets, it's easy to figure out the cardinality. Try a few examples:

For infinite sets, cardinality gets interesting, but we can say for now that the cardinality of any infinite set is $\infty$.

The "cardinality" of a set measures the number of "unique things" in it.

We say that the cardinality of $\{1, 5, 4, 1, 3\}$ is $4$.

Sometimes there are lots of unique things in a set, like the set of positive integers $\{1, 2, 3, 4, 5, 6, 7, ...\}$.

Some infinite sets are bigger than others.

Thinking about the size of an infinite set can seem tricky, but you know two sets $A$ and $B$ are the same size if you can make a "bijection" between them. This is just a fancy math word that means you can pair up every thing from $A$ to exactly one thing from $B$ in a way that every thing from $B$ gets paired up with exactly one thing from $A$.

Here's one way to think about a bijection:

Suppose you have two buckets: one contains blue marbles and the other contains red marbles. You aren't allowed to look inside the buckets, but you can take out marbles one at a time. You can only hold blue marbles in your left hand and red marbles in your right hand (one marble per hand, because you have small hands).

Now, here's a bijection from positive integers to even positive integers:

$f(x) = 2x$

That's it. Every positive integer gets mapped to exactly one even positive integer, and every even positive integer is mapped to from exactly one positive integer.

Because we can do this, we know that there are the same number of positive integers as positive even integers.