Criteria for Set Equality

So what does it mean for two things to be equal?

Intuitively, it means that they're the same. They behave the exact same way, and if you replaced (or substituted) one with the other in some expression, you'd get the same result.

Sets essentially contain things. That's what they do. If they contain exactly the same things, then how can we tell them apart?

The answer is we can't. So that seems like a good definition for equality. But how do we write that?

It turns out that this is the same as saying they are subsets of each other.

More formally, if we have two sets, $A$ and $B$. Then if $A\subset B$ then all of $A$'s elements are in $B$, and if all of $B$'s elements are in $A$, which is to say if $B\subset A$ as well, then they must contain the same elements.

So we say that $A=B$ if and only if $A\subset B$ and $B\subset A$.

It's kind of like saying if we have $a\le b$ and $b \le a$ then $a=b$.

Now that we know whether two set's are equal, we can define "proper subsets". Essentially $A$ is a proper subset of $B$ if $A\subset B$ and $A\neq B$. This is written $A\subsetneq B$.

As a reminder, any set is defined solely by the elements contained within such set. Thus, the equivalence of sets is based on their elements.

Two sets $A$ and $B$ are equal iff (if and only if) they have the same elements as each other. All elements in set $A$ must be in set $B$, and vice-versa. The vice-versa is extremely important because otherwise set $B$ might be a smaller subset of $A$, in which case the two sets do not contain the same elements. Really, if you understand how sets are defined individually, determining their equality is quite straightforward.