Reflexive Relations:

R is a reflexive relation if ≤ a, a) ∈ R, ” a ∈ A. It should be noted if there is at least one element a ∈ A such that ≤ a, a) ∉ R, then R is not reflexive.

Example:

Let 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 called a symmetric relation on A if ≤ x, y) ∈ R ⇒ ≤ y, x) ∈ R
That is, y R x whenever x R y.
It should be noted that R is symmetric iff R–1 = R
Let A = {1, 2, 3}, then R = {≤ 1, 1), ≤ 1, 3), ≤ 3, 1)} is symmetric.

Anti-symmetric Relations:

R is called a anti-symmetric relation if ≤ a, b) ∈ R  and  ≤ b, a) ∈ R ⇒ a = b
Thus, if a ≠ b then a may be related to b or b may be related to a, but never both.
Or, we have never both a R b and b R a except when a = b.

Example:

Let N be the set of natural numbers. A relation R ⊆ N x N is defined 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 called a transitive relation if ≤ a, b) ∈ R, ≤ b, c) ∈ R ⇒ ≤ a, c) ∈ R
In other words if a is related to b, b is related to c, then a is related to c.
Transitivity fails only when there exists a, b, c such that a R b, b R c but a     c.

Example:

Consider 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 interesting to note that every identity relation is reflexive but every reflexive relation need not be an identity relation. Also identity relation is reflexive, symmetric and transitive.