Cartesian product of sets of sets.

Question

If I have a set $A=\{\{a\},\{b\}\}$ and a set $B = \{c,d\}$. What does $A\times B$ equal?

Is it ${({a},c),({a},d)({b},c),({b},d)}

or is it $\{(a,c),(a,b),(b,c),(b,d)\}$?

Answer 1

$x\in A\times B$ iff there exists $p\in A$ and $q\in B$ with $x=(p,q).$

Now $p\in A\iff (p=\{a\} $ or $ p=\{b\} )$ while $q\in B\iff (q=c $ or $ q=d).$

So the members of $A\times B$ are $(\{a\},c),\; (\{a\},d),\;(\{b\},c),\;(\{b\},d).$