Does Cartesian Product and Collection of all Sets Perform a Semigroup?

Question

We know that the Cartesian Product is a binary operation. Also it is an associative operation.
We know that Cartesian Product of two set is again set, there is even closure axiom.
So I need to know does Cartesian Product and Collection of all Sets Perform a Semigroup?

Answer 1

Is the operation indeed associative? Do we have $\left(A\times B\right)\times C=A\times\left(B\times C\right)$? The sets are isomorphic in category $\mathbf{Set}$ wich means that there is bijection from one to another. But that does not mean that the sets are the same. This obstacle must somehow be 'modded' out. If that's done then you can indeed speak of a semigroup. Associativity is enough.