**Properties of Relation in a Set**

**Reflexive Relations:**Let R be a relation in a set A. then R is called a reflexive relation if(a, a)ÎR for all aÎA.

In other words, R is reflexive if
every element in A is related to itself. Thus, R is reflexive if aRa holds for
all aÎA.

As relation R is a set A is not
reflexive if there is atleast one element such that (a, a)ÏR.

Ex: Let A={1, 2, 3, 4}. Then the relation

R

_{1}={(1, 1), (2, 4), (3, 3), (4, 1), (4, 4)} in A is not reflexive since 2ÎA but (2, 2)ÏR_{1}.**Anti-Reflexive Relations:**

**Let R be a relation in a set A. Then R is called an anit-reflexive relation if (a, a)ÏR for every aÎA. i.e., a/Ra for all aÎA.**

Ex: The relation in the set of natural numbers N defined by

**(i)**‘x<y’ is anti-reflexive, since a is not less than a for any natural number a,

**(ii)**Let A={1, 2, 3, 4}. Then the relation R

_{i}={(1, 1), (2, 3), (3, 4)} is not anti-reflexive since (1, 1)ÎR.

**Symmetric Relations:**

**Let R be a relation in a set A. Then R is said to be symmetric relation if**

(a, b)ÎR Þ (b, a)ÎR.

Thus R is symmetric
if bRa holds whenever aRb holds.

A relation R is set A is not symmetric
if there exist two distinct elements a, bÎA, such that aRb, but b/Ra.

Ex:
Let R be the set of all striaght lines in aplane. The relation R is defined by
‘x is parallel to y’ is symmetric, since if a straight line a is parallelto a
straight line b, then b is also parallel to a, thus

(a, b)ÎR Þ (b,a) ÎR

**Anti-Symmetric Relation:**

**Let R be a relation in a set A, i.e., let R be a subset of A ´ A. Then R is said to be an anti-symmetric relation if**

(a, b)ÎR and (b, a)ÎR Þa=b

Thus R is anti-symmetric if a≠b then possibly (a, b)ÎR or possibly (b, a)ÎR, be never both.

A relation R is a set A is not
anti-symmetric if there exist elements a, bÎA, a≠b such that (a, b)ÎR and (b, a)ÏR.

Ex: Let A be a family of sets and let R be the
relation in A defined by ‘x is subset of y’ then R is anti-symmetric since AÍ B and BÍA ÞA=B.

**Transitive Relations:**

**Let R be a relation in a set, i.e., let R be a subset of A´A then R is said to be a transitive relation if**

(a, b)ÎR and (b, c)ÎR Þ(a, c)ÎR.

A relation R in a set is not
transitive if there exist elements a, b, cÎa, not necessarily distinct, such that (a, b)ÎR, (b, c)ÎR but (a, c)ÏR.

Ex:
Let L be the set of all straight lines in a plane and R be the relation in L
defined by ‘x’ is parallel to y’. If a is parallel to b and b is parallel to c
then obviously a is parallel to c. Thus (a, b)ÎR and (b,c)ÎRÞ(b,c)ÎR. Hence R is transitive.

**Equivalence Relations:**

Let
R be a relation in a set A. Then R is an equivalence relation in A if and only
if

**(i)**R is reflexive, i.e., for all aÎR, (a, a)Îr.

**(ii)**R is symmetric, i.e., (a, b)ÎR Þ(b,a)ÎR, for all a, bÎA.

**(iii)**R is transitive, i.e., (a, b)ÎR and (b, c)ÎR Þ(a, c)ÎR, for all a, b, cÎA.

Ex:
(i) The most trivial example of an equivalence relation is that of ‘equality’
for any elements in any set.

**I.**A=a, i.e., reflexive

**II.**a=bÞ b=a i.e., symmetric

**III.**a=b and b=cÞa=c, i.e., transitive.

(ii)
Let R be the relation in the real numbers defined by ‘x£y’. Then

**I.**x £ x i.e.(x,x)ÎR, i.e., R is reflexive.

**II.**Let x£ y but y not £ x i.e., (x, y)ÎR but not necessarily, (y, x)ÎR i.e., R is not symmetric. Thus R is not an equivalent relation.

**Partial Ordered Relation:**

**A relation ‘R’ on A such that ‘R’ is reflexive, anti symmetric and transitive is called partial order relation on ‘A’.**

**Note:**

**(i)**Union of ‘2’ Reflexive or symmetric relation is a reflexive or symmetric relation.

**(ii)**Union of ‘2’ transitive relations need not be transtitive.

**(iii)**Intersection of ‘2’ equivalance relation is an equivalence relation.