Properties of Relations in a Set:

Reflexive Relations:

R is a reflexive relation if (a, a) ∈ R, ∀ a ∈ A. It could be noticed if there is at least one member a ∈ A such that (a, a) ∉ R, then R is not reflexive.

Problem:       Suppose A = {1, 2, 3, 4, 5}

                        R = {(1, 1), (3, 2), (4, 2), (4, 4), (5, 2), (5, 5)} is not reflexive because 3 ∈ A  and (3, 3) ∉ R.

                        R = {(1, 1), (3, 2), (2, 2), (3, 3), (4, 1), (4, 4), (5, 5)} is reflexive since  (a, a) ∈ R, " a ∈ A.

Symmetric Relations:

R is known as a symmetric relation on A if (x, y) ∈ R => (y, x) ∈ R

That is, y R x whenever x R y.

It could be noticed that R is symmetric iff R^-1 = R

Suppose A = {1, 2, 3}, then R = {(1, 1), (1, 3), (3, 1)} is symmetric.

Anti-symmetric Relations:

R is known as a anti-symmetric relation if (a, b) ∈ R  and  (b, a) ∈ R => a = b

Therefore, if a ≠ b then a can be belongs to b or b can be belongs to a, but never both. Or, we have never both a R b and b R a apart from when a = b.

Problem:       Let N be the set of natural integer numbers. A relation R ⊆ N x N is described by

                        x R y iff x divides y (i.e. x/y)

                        Then x R y, y R x => x divides y, y divides x => x = y

Transitive Relations:

R is known as a transitive relation if (a, b) ∈ R, (b, c) ∈ R Þ (a, c) ∈ R

In other terms if a belongs to b, b belongs to c, then a belongs to c. 

Transitivity be unsuccessful only when there exists a, b, c such that a R b, b R c but a c.

Example:      Suppose the set A = {1, 2, 3} and the relation

                        R₁ = {(1, 2), (1, 3)}

                        R₂ = {(1, 2)}

                        R₃ = {(1, 1)}

                        R₄ = {(1, 2), (2, 1), (1, 1)}

Then R₁, R₂ and R₃ transitive while R₄ is not transitive since in R₄, (2, 1) ∈ R₄,  (1, 2) ∈ R₄ but (2, 2) ∉ R₄

Note:

• It is remarkable to notice that every identity relation is reflexive but every reflexive relation used not be an identity relation. Also identity relation is reflexive, transitive and symmetric.

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