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**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

R_{1} = {≤ 1, 2), ≤ 1, 3)}

R_{2} = {≤ 1, 2)}

R_{3} = {≤ 1, 1)}

R_{4} = {≤ 1, 2), ≤ 2, 1), ≤ 1, 1)}

Then R_{1}, R_{2} and R_{3} transitive while R_{4} is not transitive since in R_{4}, ≤ 2, 1) ∈ R_{4}, ≤ 1, 2) ∈ R_{4} but ≤ 2, 2) ∉ R_{4}

** 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.

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