Suppose A, B and C are sets such that #A=#B. Is it true that #(A x C) =#(B x C)? If it is, prove that.

Question

I'm sure that this is true, because A and B have the same cardinality (number of elements in the set). So there has to be the same number of ordered pairs with C in both cases. But I'm not sure how to exactly prove it since we just started with cardinality.

Answer 1

You just need a bijection between $A \times C$ and $B \times C$ given a bijection between $A$ and $B$.

Let $f: A \rightarrow B$ be a bijection. Then define $g: A \times C \rightarrow B \times C$ given by $g(a,c)=(f(a),c).$

Answer 2

Ac= Ac ( for any value)

Now given that a=b Now we can replace the right hand side "a" of this given equation with its value B, Then equation will be

AC = BC (proved)