Relations

Cardinal number of Cartesian product:
The cardinal number of the Cartesian product A\( \times \)B denoted by n(AB), is the product of the cardinal numbers of the sets A  and B, that is , n(A\( \times \)B) = n(A). n(B) where n(A) = Cardinal number of A, n(B) = number of elements of B. 
n(A\( \times \)B) = n (B\( \times \)A).

Example: Find n(A\( \times \)B) if A = {1, 2, 3, 4, 5} and \( B = \left\{ {x/2 \leqslant x \leqslant 7,x \in N} \right\} \)
Solution: Here A = {1, 2, 3, 4, 5}. So n(A) = 5  
\( B = \left\{ {x/2 \leqslant x \leqslant 7,x \in N} \right\} \)= {2, 3, 4, 5, 6, 7}. So n(B) = 6
     n(A\( \times \)B) = n(A).n(B) = 5.6 = 30.

Note: If n(A) = k the number of subsets of A is 2^k, i.e.., 2^n(A)
    So, the number of subsets of A\( \times \)B is 2^{n(A x B)}, i, e.., 2^n(A).n(B)

Graphical representation (lattice diagram) of Cartesian product of two sets:
  

Let A = {1, 2} and B = {2, 3} then A\( \times \)B = {(1, 2),(1, 3),(2, 2), (2, 3)} 
                                                  B\( \times \)A = {(2, 1),(2, 2),(3, 1), (3, 2)}
Take two mutually perpendicular lines OX and OY. Using a scale represent the number 1 and 2 (members of A) one OX, and the numbers 2 and 3 (members of B) on OY. Draw lines parallel to OY Through the points representing 1 and 2 (members of A). Again draw lines parallel to OX through the point 2 and 3 (members of B) write the ordered pairs of  numbers at the points of intersection of the lines drawn similarly, we can represent B\( \times \)A graphically as shown in the figure.

Order triplet: If A, B and C are three sets, then (a, b, c) where a\( \in \)A, b\( \in \)B and c\( \in \)C, is called ordered triplet.