Relation

Ordered Pairs & Cartesian Products

1. Ordered Pairs
→ A pair having one element as the first and the other as the second can be understood as Ordered Pairs. An ordered having 'a' as the first element and 'b' as the second element is denoted as (a, b). 
     Two ordered pairs are said to be equal if the corresponding elements of the pairs are equal i.e., if (a, b) and (c, d) are two ordered pairs then the ordered pairs are equal only if a=c & b=d. An ordered pair (a, b) ≠ (b, a) unless a=b.
     i.e.,
            (1, 2)=(1, 2)
            (1, 2)≠(2, 1)
            (1, 1)=(1, 1)

2. Cartesian Product
→ The set of all ordered pairs (a, b) such a ∈ A and b ∈ B where A and B are two non-empty sets is called as Cartesian Product of A and B. The cartesian product of A and B is denoted by  A × B. 

      The cartesian product of A = {1, 2, 3} and B = {5, 6} is
            (i)B x A = {5, 6) x {1, 2, 3}
            = {(5, 1), (5, 2), (5, 3), (6, 1), (6, 2), (6, 3)}

            (ii)A x B =  {1, 2, 3} x {5, 6)
            = {(1,5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6)}

            (iii)A x A = {1, 2, 3} x {1, 2, 3}
            = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

            (iv)B x B = {5, 6} x {5, 6}
            = {(5, 5), (5, 6), (6, 5), (6, 6)}
    ➝In general, A x B ≠ B x A.
    ➝If m is the number of elements in A and n is the number of elements in B, then the         number of elements in A x B or B x A is mn.
    ➝The Cartesian product of A with itself i.e., A x A is also known as Cartesian product on A.

1.Relation
→A Relation between two sets is a collection of ordered pairs containing one element from given two sets which satisfies the given relation. In simple words, Relation is a subset of Cartesian product.

➣Inverse Relation: The inverse relation for the relation from A to B is given by the relation            from B to A.
    If {(1, 2)(1, 4)(3, 2)(3, 4)} is a relation from A to B where A={1, 3} & B={2, 4} then the                    inverse of given relation is {(2, 1)(4, 1)(2, 3)(4, 2)} which is the relation from B to A.