Cartesian Representation
Cartesian Representation of a Relation is a part of cartesian product. Given two sets A and B. Then, all possible ordered pairs(x, y) obtained such that 
and 
is called the Cartesian product of the sets A and B. It is denoted by A x B and is read as 'A cross B'.Given two sets A and B. Then, all possible ordered pairs (x, y) obtained such that and is called the Cartesian product of the sets A and B. It is denoted by A x B and is read as 'A cross B'.
A x B = {(x,y): x 
A, y B}
Similarly B x A = {(x, y) : x B, y A}
Example: If A = {3, 5}, B = {2, 4, 6}, write the Cartesian product (i) A x B (ii) B x A
A x B = {(3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)}
B x A = {(2, 3), (2, 5), (4, 3), (4, 5), (6, 3), (6, 5)}
Note:
(i) A x B 
B x A unless A = B
(ii) n(A x B) = n(A) x n(B)
(i) We can plot the ordered pairs in A x B by taking the first element along the X-axis and the second element along the Y-axis in the Cartesian plane. This diagram is called Lattice. Each ordered pair is marked by a point called the LATTICE point.
A x B = {(3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)}
(ii) Similarly we can plot the ordered pairs in B x A.
B x A = {(2, 3), (2, 5), (4, 3), (4, 5), (6, 3), (6, 5)}
The plural of lattice is 'lattices'.
It is named after the mathematician George Papy. The elements of one set are placed in one circle and the elements of the other set are placed in the second circle. Arrows indicate the passing of elements of set A to set B or set B to set A.
(i) A x B = {(3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)}
(ii) B x A = {(2, 3), (2, 5), (4, 3), (4, 5), (6, 3), (6, 5)}
If A = {1, 2, 3}, B = {3, 6}
Find (i) A x B (ii) A x A (iii) B x B.
Hence draw Papy graph in each case.
(i) A x B
A x B = {(1, 3), (1, 6), (2, 3), (2, 6), (3, 3), (3, 6)}
(ii) A x A
A x A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
(iii) B x B
B x B = {(3, 3), (3, 6), (6, 3), (6, 6)}
If A = {1, 2}, B = {2, 3}, C = {3, 4}
Find (i) (A x B) 
(A x C) (ii) (A x B) (B x C)
A x B = {(1, 2), (1, 3), (2, 2), (2, 3)}
A x C = {(1, 3), (1, 4), (2, 3), (2, 4)}
(A x B) (A x C) = {(1, 3), (2, 3)}
B x C = {(2, 3), (2, 4), (3, 3), (3, 4)}
(A x B) 
(B x C) = {(1, 2), (1, 3), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4)}
