Cardinality of two sets cross-multiplied

Question

Let $A$ and $B$ be sets. Prove that $ \#(A \times B) = \#(B \times A)$.

What I have done:

There exist an element $m$ in $A$ such that the element also exists in $B$. If $\#A = \#B$, then $\#B = \#A$. Any two sets cross-multiplied must have equal sets of elements or else you cannot cross-multiply.

Also, there exist elements $(1,2,...n)\in A$ that is also $\in B$.

I need help using what I have, or maybe I shouldn't use this information for this proof...

Answer 1

The map $f:A\times B\to B\times A,(a,b)\to (b,a)$ is 1-1 and surjective.Let $(b',a')=(b,a)$ in $B\times A$. Then $b'=b$ and $a'=a$. This means that $(a',b')=(a,b)$ in $A\times B$ (1-1).

Also if $(b,a)\in B\times A$ then $f((a,b))=(b,a)$ (surjection).

Answer 2

Also, for a more graphic proof, if $A=\{a_1,\ldots,a_n\}$ and $B=\{b_1,\ldots,b_m\}$ are finite, then

$A\times B=\{(a_1,b_1),\ldots,(a_1,b_m),(a_2,b_1),\ldots,(a_2,b_m),\ldots,(a_n,a_1),\ldots,(a_n,b_m)\}$,

which are $mn$ elements and a similar enumeration shows that $B\times A$ also has $mn$ elements.