Are these two products equal? Solution By the definition of the cartesian product, P × Q = {( a , r ), ( b , r ), ( c , r )} and Q × P = {( r , a ), ( r , b ), ( r , c )} Since, by the definition of equality of ordered pairs, the pair ( a , r ) is not equal to the pair ( r , a ), we conclude that P × Q ≠ Q × P. However, the number of elements in each set will be the same. Example Let A = { , , }, B = { , } and C = { , , }.
Find A × (B ∩ C) (A × B) ∩ (A × C) (iii) A × (B ∪ C) (iv) (A × B) ∪ (A × C) Solution (i) By the definition of the intersection of two sets, (B ∩ C) = { }. Therefore, A × (B ∩ C) = {( , ), ( , ), ( , )}. Now (A × B) = {( , ), ( , ), ( , ), ( , ), ( , ), ( , )} and (A × C) = {( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )} Therefore, (A × B) ∩ (A × C) = {( , ), ( , ), ( , )}. (iii) Since, (B ∪ C) = { , , , }, we have A × ( B ∪ C) = {( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )}.
(iv) Using the sets A × B and A × C from part (ii) above, we obtain (A × B) ∪ (A × C) = {( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )}. RELATIONS AND FUNCTIONS Example If P = { , }, form the set P × P × P. Solution We have, P × P × P = {( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , ), ( , , )}. Example If R is the set of all real numbers, what do the cartesian products R × R and R × R × R represent?
Solution The Cartesian product R × R represents the set R × R