Can we compare $2$ sets? I've learned that you can add, subtract or multiply a constant to a set of numbers. Infact one can add $2$ sets as well.
But what does it mean to compare $2$ sets, for instance, $A>B$ or $A≥B$ what do those two even mean?
Question 2 : Is there such a thing as multiplying $2$ sets? Not the Cartesian product, you know, just simple multiplication?
 A: In general, one actually cannot add two sets in the way you describe. We only get notions of combining/comparing two sets in an "algebraic" way when we know ahead of time that the sets consist of elements of some pre-existing algebraic structure (e.g. they're sets of real numbers).
Once we do that, this algebraic structure can be "lifted" to the level of sets. For example:


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*If we're looking at sets of real numbers, we can "add" and "multiply" them in a natural way: $$A+B=\{a+b: a\in A,b\in B\},\quad A\cdot B=\{a\cdot b: a\in A,b\in B\}.$$ Similarly we can deal with exponentiation, and etc.


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*Sometimes these operations are useful enough to get special names - e.g. the above notion of "addition" on sets of real (or similar) numbers, or more generally vectors of such numbers, is called the Minkowski sum.


*Now, comparing sets is a "yes/no" affair: we're not generating a new set, we're just defining a relation on sets. The most natural way to do this is to ask whether the given relation at the level of numbers always holds with respect to the elements of the set; e.g. we might say $A\le B$ if every element of $A$ is $\le$ every element of $B$ (that is, for each $a\in A$ and $b\in B$ we have $a\le b$).

Below I'm going to write expressions like "$A-B$" to refer to the operations on sets gotten from the above procedure; this isn't really done, and in general "$A-B$" would be interpreted as the set difference $\{x: x\in A, x\not\in B\}$, but I think this notation will help make this answer clear.

However, we must note the following issues:


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*First, as observed above these "algebraic" operations only make sense when we're working with subsets of a given algebraic structure.  For general sets, there's no notion of addition analogous to the above.

*Second, even when we do have some algebraic structure to play with, its basic properties might change drastically when we move from elements (e.g. numbers) to sets. For example:


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*We don't in general have $(A-A)+A=A$, even though $(a-a)+a=a$ is always true for real numbers: e.g. taking $A=\{1,2\}$, we get $$A-A=\{1-1,1-2,2-1,2-2\}=\{-1,0,1\}$$ and so $$A-A+A=\{-1+1,0+1,1+1, -1+2,0+2,1+2\}=\{-1,0,1,2,3\}.$$

*Sets of real numbers aren't linearly ordered in the notion above, in contrast with individual real numbers: e.g. if $A=\{1,2\}$ and $B=\{0, 3\}$, then $A\not\le B$ and $B\not\le A$.


*Finally, the "algebraic operations lifted from elements" have nothing to do with the general set-theoretic operations we can apply to an arbitrary set (like powerset, Cartesian product, symmetric difference, etc.) This can often lead to notational issues when mixing the two types of operation, and one has to be very careful here.
