Compositions of Relations:

Consider R ⊆ A x B, S ⊆ B x C be two relations. Then compositions of the relations S and R shown by So R ⊆ A x C and is described by (a, c) ⊆ (S o R) iff ∃ b ∈ B such that (a, b) ∈ R, (b, c) ∈ S.

Example:      consider A = {1, 2, 3}, B = {a, b, c, d}, C = {α, β, γ}

                        R(⊆ A x B) = {(1, a), (1, c), (2, d)}

                        S (⊆ B x C) = {(a, α), (a, β), (c, γ)}

                        Then S o R(⊆ A x C) = {(1, a), (1, g), (1, b)}

One could be careful in calculating the relation R o S. Actually S o R initiate with R and R o S starts with S. In common S o R ≠ R o S. Also (S o R)^-1 = R^-1 o S^-1, called as reversal principal

Inverse Relation:

Consider R ⊆ A x B be a relation from A to B. Then inverse relation R-1 ⊆ B x A is described by R^-1 = {(b, a): (a, b) ∈ R, a ∈ A, b ∈ B}. It is obvious that

• a R b <=> b R^-1 a
• dom R^-1 = range R and range R^-1 = dom R
• (R^-1)^-1 = R

Problem:       Let A = {1, 2, 3, 4}, B = {a, b, c} and R = {(1, a), (1, c), (2, a)}. Then

                        (i) dom R = {1, 2}, range R = {a, c}

                        (ii) R^-1 = {(a, 1), (c, 1), (a, 2)}

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