Exercise 2.51. Let A, C C X and B, D CY. (1) Show that Ax BC X x Y. (2) Prove or disprove: (X \ A) × (Y \B) = (X × Y)\ (A x B). (If disproved, what does hold?) (3) Show that (A x B)n (C × D) = (AnC) x (Bn D). (4) Can we replace n by U in the above statement? (If not, what does hold?) %3D

Note: As per bartleby instruction when more then one question is given only one has to be answered. Please upload other separately.

(1) The aim is to show that $A\times B\subset X\times Y$

Given that, $A,C\subset X, B,D\subset Y$.

Now, it is known that,

$A\times B=\left\{\left(a,b\right)|a\in A, b\in B\right\}$

It is given that,$A\subset X, B\subset Y$.

So, $A\times B\subset X\times Y$.

(2) The objective is to show that, $\left(X\backslash A\right)\times \left(Y\backslash B\right)=\left(X\times Y\right)\backslash \left(A\times B\right)$

This can be written as, 

$\begin{array}{rcl}X\backslash A\times \left(Y\backslash B\right)& =& \left\{\left(x,y\right)|x\in X and x\notin A, y\in Y and y\notin B\right\}\\ & =& \left\{\left(x,y\right)|x,y\in X,Y and x,y\notin A,B\right\}\end{array}$

Since $x\notin A,y\notin B$ imply that, $(x,y)\notin \left(A\times B\right)$.

So, $\left(X\backslash A\right)\times \left(Y\backslash B\right)=\left(X\times Y\right)\backslash \left(A\times B\right)$.