Let A and B be finite sets Prove that the Cartesian product

Let A and B be finite sets. Prove that the Cartesian product A Times B is finite.

A set is finite if we can put it in one to one correspondence with some finite set

Let, m=|A| ,n=|B|

So we can label elements of A as:a_1,...a_m

And of B as:b_1,..,b_n

Consider the matrix C of size: mxn with mn entries.

C_{ij}=j+n*(i-1)

So, first row entries go from: 1 to n

Second row form :n+1... 2n and so on

So all entries are different from on another

Let set ofall entries of C be denoted by X

We define the map from AxB to X as

f(AxB)-->X

f((a_i,b_j))=X_{i,j}

f((a_i,b_j))=f((a_k,b_r))

Means

X_{i,j}=X_{k,r}

j+n(i-1)=r+n(k-1)

(j-r)=n(k-i)

But, 1<=j,r<=n

Hence, j=r, k=i

Hence f is injective

For any X_{ij} there is (a_i,b_j) so that

f((a_i,b_j))=X_{ij}

Hence f is onto

So AxB is finite