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.
 A: (I assume that $U$ denotes some universal set, or universe of discourse: simply put, a set which contains everything currently under discussion.)
Well, $A$ union everything must be at least as big as everything.  But everything is, well, everything, so no set can be bigger than everything.  Therefore $A$ union everything must be everything.
A: The right circle can't be $U$, because there are points not contained in the right circle. Remember that $U$ is the universal set: all points (in the domain of discussion) are a member of $U$.
If some set is missing any point, then that set is not $U$.
A: 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 $A$.
A: The set equalities are explained by logic. 
Let's call T ( or "truth") the proposition that is equivalent to all tautologies ( that is, all logically/ necessarily true propositions). 
Let's calll F ( or "falsity) the proposition that is equivalent to all antilogies ( that is, all logically/ necessarily false  propositions). 
You can verify, using a truth table that , for all proposition P 
(1) P OR F is equivalent to P 
(2) P & F is equivalent to F 
(3) P OR T is equivalent to T. 
(4) P & T is equivalent to P 
Let's explain (1). The OR operator requires, to make a true statement, that 1 at least of the two proposiitons be true. Since the second, being "F"  is ( by definition) always false, everything depends on the truth value of the first : P. If P is true, it is a sufficient condition for (P OR F) to be true. If P is false, it is a sufficient condition for (P OR F) to be false. You can see that P and (P OR F) always have the same truth value: they are equivalent. 
With this in mind, you can understand your set equalities, remembering the logical definition of each set theoretic operation. 
For example 
A Union EmptySet is , by definition the set of all x such that : 
     (x belongs to A) OR (x belongs to the empty set). 

But "x belongs to the empty set" is a logical impossibility: it can be replaced by the constant F ( = the false) 
So, our set is the set of all x such that : 
      ( x belongs to A) OR F  

And, as we have seen, this OR sentence  is equivalent  to 
      " x belongs to A ". 

But " the set of all x such that x belongs to the set A "is nothing else but the set A itself!
Therefore : The set A Union EmptySet is nothing else but the set A itself. 
To prove your set equalities, rephrase them using the logical definition of set operations and use : 


*

*the constant T for " x belongs to U " ( U being the universal set) ; for " x belongs to U " is logically/necessarily true ( every element x being by definition an element of the universal set  "U") 

*the constant F for " x belongs to the EmptySet" 
Remark. It is somewhat misleading to say that the universal set U is everything. For all the sets you are dealing with are something ( even the empty set, which is not not nothing, but a set); but this sets are not elements of the universal set, they are subsets of the universal set. So U ( the universe) is, so to say " every elements" , but not really  " everything". 
