1- Let A={a,b} and B={a,b,c,d,e}. I know that B is superset of A. But it seems a universal set as well.
2- If A={a,b}, B= {a,b,c,d,e} and C= {c,d}. B is still a superset of A and C. Is it also a universal set of A and C or is it necessary that Universal set for A and C be B or U= {a,b,c,d}.
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The definition of universal set is the set containing all objects or elements and of which all other sets are subsets.
Question 1) If we look at $B = \{a,b,c,d,e \}$, we could say the $B \subseteq U$ if the whole universe of elements are $U = \{ a,b,c,d,e \}$. So if $a,b,c,d,e$ are all the elements in this universe then, yes, $B$ is a equal to the universal set.
Question 2) Let's think of what it means to be a superset. A superset is a set of elements containing all of the elements of another set. So we see that $B$ is a superset of $A$ and $C$ because $B$ has all the elements inside of $A$ and $C$. Thus, $B \supset A$ and $B \supset C$, and if $B$ has all the elements in the universe, then $B$ is equal to the universal set, meaning $B \subseteq U$.
EDIT:
When saying $U = \{a,b,c,d \}$ do you mean for $B = \{ a,b,c,d \}$? No set can have more elements than the universal set.
I think you are mistakenly thinking there is some kind of rule or definition to make the superset of a set or group of sets, and that you can have a universal set of some sets and another universal set of other sets.
That's not what these mean.
A superset of a set means nothing more than a set that contains all the elements. So for example if $A = \{a,b\}$ then $M = \{a,b,c\}$ is a superset. And so is $N=\{a,b,f,q\}$ as is $J = \{a,b,w, \phi, \text{Babar the elephant}, apple\}$. There is no one superset.
Now if $A = \{a,b\}$ and $C= \{b,c\}$ then $\{a,b,f,w\}$ is a superset of $A$ but not of $C$. And $\{b,c, e, m\}$ is a superset of $C$ but not of $A$. But $\{a,b,c, k, z\}$ is a superset of both.
Now, it's worth noting that $A \cup C$ is the set that contains all the elements of both $A$ and $C$ (and no other elements) so $A\cup C = \{a,b,c\}$. It may be worth noting that that would mean $A\cup C$ is the smallest possible superset of both $A$ and $C$. But it not the only superset of both $A\cup C$.... (assuming there are more things in your universe than just $a,b,c$.)
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So what is the universal set? Well, for every problem there is exactly one universal set and it is the set that has EVERYTHING you are allowed to use.
You'll notice in an example above I put some silly things in my set $J$. I put in $\phi$ (are we allowed to use greek letters?) and I put in Babar the elephant (are we allowed to use fictional children's book characters) and I put in $apple$ (are we allowed to use fruit?)
Are they allowed? I don't know. It depends on how the problem was set up and what we are allowed to use.
And that is all that the universal set is. It is the set of everything we are allowed to use. And there is only one for every problem.
So in the problems you mention I see you use the elements $a,b,c,d,e$. So the universal set must include those (otherwise we couldn't use them). But what else does it include?
I don't know. You didn't tell me. If $a,b,c,d,e$ is everything we are allowed to use then the universal set is $U=\{a,b,c,d,e\}$. But if we are allowed to use $f$ (even if we don't use it) the $f$ must be in the universal set.
Often the Universal set isn't specifically spelled out and it is only implied. Maybe it is assumed you are taking sets of real numbers and it is assumed the universal set is $\mathbb R$.
EDIT: The author is defining the universal set to be A set that takes all elements mentioned into account. As in both example 1 and 2, the only elements mentioned are $a,b,c,d,e$ the universal set is $\{a,b,c,d,e\}$ which happens to by set $B$.
In example 1) $B$ is a superset of $A$. The other possible supersets are $\{a,b\}$ (Every set is a superset of itself), $\{a,b,c\},\{a,b,d\},\{a,b,e\},\{a,b,c,d\},\{a,b,c,e\},\{a,b,d,e\}$ and $\{a,b,c,d,e,\}$ (the universal set is always a superset of every set)
In example 2) $B$ is still a superset of $A$ both and of $C$. It is the universal set so it is a superset of every set. It is not the universal set of $A$ and $C$. There is no such thing as a universal set for individual sets. There is only ONE universal set for EVERYTHING. ANd that is still $\{a,b,c,d,e\}$. It's true that neither $A$ nor $C$ has element $e$ but $B$ does. The one universal set for everything must be a superset of all sets; not just the two you are talking about.
$A\cup C = \{a,b,c,d\}$ is the smallest set that is a superset of both $A$ and of $C$. But the universal set, by definition, would be the biggest superset of bot $A$ and of $B$.