Suppose that A,B,C,D are sets with A≈C and B≈D. Prove that AxB≈DxC?

I assume that your notation means isomorphic. In this case, let f:A-%26gt;C and g:B-%26gt;D be the isomorphisms (that is, f an g are both one to one and onto). Define a new function h:AxB-%26gt;CxD by setting h(a,b)=(f(a),g(b)). Now we just need to check that this the desired isomophism.





one-to-one: if h(a1,b1)=h(a2,b2) then (f(a1),g(b1))=(f(a2),g(b2)). So, we must have that


f(a1)=f(a2) and g(b1)=g(b2), but f and g are one-to-one which means that a1=a2 and b1=b2 giving (a1,b1)=(a2,b2).





onto: take (c,d) in CxD. Again, f and g are onto, so there exists a in A and b in B such that f(a)=c and g(b)=d. Thus, h(a,b)=(c,d) as desired.





We have checked that h is one-to-one and onto, so it is an isomorphism.