Identity Law - Set Theory I'm trying to wrap my head around the Identity Law, but I'm having some trouble.
My lecture slides say:
$$
A \cup \varnothing = A
$$
I can understand this one. $A$ union nothing is still $A$. In the same way that $1 + 0$ is still $1$.
However, it goes on to say:
$$
A \cup U = U
$$
I don't see how this is possible. How can $A\cup U = U$?
http://upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/150px-Venn0111.svg.png
If this image represents the result of $A\cup U$, where $A$ is the left circle, $U$ is the right circle, how can the UNION of both sets EQUAL the RIGHT set? I don't see how that is possible?
Can soemone please explain to me how this is possible? Thanks.
$$
    A \cap\varnothing=\varnothing,\\
    A \cap U = A
$$
Some further examples from the slides. I really don't understand these, either. It must be something simple, but I'm just not seeing it.

Well $U$ is "the universe of discourse" -- it contains everything we'd like to talk about. In particular, all elements of $A$ are also in $U$.
In the "circles" representation, you can think of $U$ as the paper on which we draw circles to indicate sets like $A$.
