Question

Some elements of A × B are (a,x) (c,y) (d,z).
If A = { a,b,c,d}. Find the remaining elements of A×B such that n(A×B) is least.

Answer 1

Answer:

Let A={a,b,c,d,e}

and B={1,2,3,4}

A×B=(a,1),(a,2),(a,3),(a,4),(b,1),(b,2),(b,3),(b,4),(c,1),(c,2),(c,3),(c,4),(d,1),(d,2),(d,3),(d,4),(e,1),(e,2),(e,3),(e,4)

∴n(A×B)=n(A)n(B)=20

We know that (A×B)∩(B×A)=(A∩B)×(B∩A)

So,(A×B)∩(B×A)=(A∩B)×(B∩A)

Given:Common elements of A and B=3

∴n(A∩B)=3

So,n[(A∩B)×(B∩A)]=n(A∩B)n(B∩A)=3×3=9 elements

∴n[(A∩B)×(B∩A)]=9 elements