Relations

1. Cartesian product (or cross product) of two sets
    The Cartesian product of two sets A and B, denoted by A\( \times \)B (read A cross B), is the set of all possible ordered pairs (a, b) where a\( \in \)A, b\( \in \)B,
    In set – builder form A\( \times \)B = {(a, b)/ a\( \in \)\( \in \)A, b\( \in \)B}
    Similarly, the Cartesian product B\( \times \)A of the sets B and A is the set 
    B\( \times \)A = {(b, a)/ b\( \in \)B, a\( \in \)A}
    Thus, (a, b)\( \in \)A\( \times \)B but (b, a)\( \notin \)A\( \times \)B.
    
    (b, a)\( \in \)B\( \times \)A but (a, b)\( \notin \)B\( \times \)A unless a and b both belong to A and B
Example :- Find the Cartesian products A\( \times \)B and B\( \times \)A of the sets A and B where  {1, 2, 3},  B = (x/x² = 1}
Solution:
    Writing the set B in tabular form, B = {- 1, 1}
    Thus we have A = {1, 2, 3}, B = {- 1, 1}
     \( \therefore \) By definitions, A  B = {(1, -1), (1, 1), (2, -1) (2, 1),(3, - 1),(3, 1)}
     B\( \times \)A = {(-1, 1), (-1, 2), (-1, 3) (1, 1),(1, 2),(1, 3)}
Method of writing the set A\( \times \)B in roster form
Step 1: Write the Sets A and B in roster form.
Step 2: Take elements of A as the first member and all the elements of B as second member one by one, and form ordered pairs.
Step 3: Repeat step 2 for all the elements of A
Step 4: Write a set with all the ordered pair obtained using steps 2 and 3

This diagram is known as arrow diagram A\( \times \)B.
The arrow diagram of A\( \times \)B in the above example is given below.

Note: If A or B is a null set \( \emptyset \), A\( \times \)B = \( \emptyset \)
\( A \ne B,\,A \times B \ne B \times A \)

Cartesian product A\( \times \)A
    The Cartesian product of a set A with itself written as A \( \times \)A is the set of all ordered pairs (a, b) where a A and b\( \in \)A. For example, let A = {1, 2}

Then A\( \times \)A = {(1, 1), (1, 2), (2, 1), (2, 2)}