, b , b , b } (Fig . ) . A × B = {( a , b ), ( a , b ), ( a , b ), ( a , b ), ( a , b ), ( a , b ), ( a , b ), ( a , b )}. The ordered pairs thus formed can represent the position of points in the plane if A and B are subsets of the set of real numbers and it is obvious that the point in the position ( a , b ) will be distinct from the point in the position ( b , a ).
Remarks Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal. DL MP KA Fig . Fig . MATHEMATICS If there are p elements in A and q elements in B, then there will be pq elements in A × B, i.e., if n (A) = p and n (B) = q, then n (A × B) = pq .
(iii) If A and B are non-empty sets and either A or B is an infinite set, then so is A × B. (iv) A × A × A = {( a , b , c ) : a , b , c ∈ A}. Here ( a , b , c ) is called an ordered triplet . Example If ( x + , y – ) = ( , ), find the values of x and y .
Solution Since the ordered pairs are equal, the corresponding elements are equal. Therefore x + = and y – = . Solving we get x = and y = . Example If P = { a , b , c } and Q = { r }, form the sets P × Q and Q × P.