Question

If P = {t‚ o‚ p} and Q = {p‚ o‚ t}‚ find P × Q and Q × P.
Is P × Q = Q × P? Give reason.

Answer 1

Given : P = { t, o, p } and Q = { p, o, t }

To find : Find sets P × Q and Q × P , and find whether P × Q = Q × P

Solution :

Cartesian product of two sets is given by, ( x, y ) such that x ∈ P and y ∈ Q.

From the defination of cartesian product,

P × Q = { (t, p), (t, o), (t, t), (o, p), (o, o), (o, t), (p, p), (p, o), (p, t) } _______(1)

Q × P = { (p, t), (p, o), (p, p), (o, t), (o, o), (o, p), (t, t), (t, o), (t,p) } _________(2)

From (1) and (2)

It is clear that P × Q = Q × P because all elements of P × Q are equal to the elements of Q × P. ( Order of the element doesn't matter ).

Note : If we are given two sets A and B then it is not always true that P × Q = Q × P. It can be equal in some specific cases but not for all.