Relation As A Subset Of The Cartesian Product

If A & B are any two non-empty sets, then any subset of A X B is defined as a relation from A & B

For example, suppose A = {1, 2, 3}, and B = {1, 2, 3, 4}. Then {(2, 3), (2, 4), (1, 3)}, is a relation in in A X B. Many more relations (subsets) can be selected similarly at random from our product set A X B.

Domain, Co-domain, and Range of a Relation

Let R be a relation from set A to set B. Then the set of first entries of the ordered pairs in R is called the domain.

Thus Domain = {a ǀ (a, b) ∈ R}

Similarly the set of second entries of the ordered pairs in R is called the range.

Thus Range = {b ǀ (a, b) ∈ R}

The second set B is called the co-domain of R.

For Example.If A = {16, 25, 36, 49}, B = {1, 4, 5, 6} and R be the relation “is square of ” from A to B then,

     R = {(a, b), ǀ a = b², a ∈ A, b ∈ B}

     R = {(16, 4), (25, 5), (36, 6)}, Domain of R = {16, 25, 36}

Range of R = {4, 5, 6}, co-domain of R = {1, 4, 5, 6}

There are some important note to be remembered which is given below -

Remember.1) The number of relations that are possible from a set A of m elements to another set B of n elements.

So, number of elements in A = m, number of elements in B = n,

Number of elements in A X B = mn,

So, number of subsets of A X B = 2ᵐⁿ

Since every sub-set of A X B is a relation from A to B, therefore 2ᵐⁿ relations are possible from A to B.

Remember.2) If R₁ and R₂ are two relations from A to B, then R₁ ∪ R₂, R₁ ∩ R₂, R₁ - R₂, are also relations from A to B.

Remember.3) Since ϕ ⊆ A X B, therefore, ϕ is a relation from A to B. Also Dom ϕ = ϕ and Range ϕ = ϕ.