Symmetric Relation

SYMMETRIC RELATION

Let P be a set on which the relation R is defined. Then R is said to be a symmetric relation, if (p, q) ∈ R ⇒ (q, p) ∈ R, that is, pRq ⇒ qRp for all (p, q) ∈ R.

e.g- the set P of natural numbers. If a relation P be defined by “x + y = 5”, then this relation is symmetric in P, for

p + q = 5 ⇒ q + p = 5

But in the set P of natural numbers if the relation R be defined as ‘x is a divisor of y’, then the relation R is not symmetric as 4R8 does not imply 8R4  for, 4 divides 8 but 8 does not divide 4.

Solved example on symmetric relation on set:

1. A relation R is defined on the set Z by {p R q if p – q is divisible by 5} for p, q ∈ Z. Examine if R is a symmetric relation on Z.

Solution:

Let p, q ∈ Z such that pRq hold. Then p – q is divisible by 5, i.e. p – q =5z and therefore q – p =5z hence q - p is divisible by 5.

Thus, pRq ⇒ qRp and therefore R is symmetric.

2. A relation R is defined on the set Z (set of all integers) by {pRq if and only if 2p + 3q is divisible by 5}, for all p, q ∈ Z. Examine if R is a symmetric relation on Z.

Solution:

Let p, q ∈ Z such that pRq holds i.e., 2p + 3q = 5z, which is divisible by 5. Now, 2q + 3p = 5p – 2p + 5q – 3q = 5(p + q) – (2p + 3q) is also divisible by 5.

Therefore pRq implies qRp for all p,q in Z i.e. R is symmetric.

3. Let R be a relation on Q, defined by R = {(p, q) : p, q ∈ Q and p – q ∈ Z}. Show that R is Symmetric relation.

Solution:

Given R = {(p, q) : p, q ∈ Q, and p – q ∈ Z}.

Let (p,q) ∈ R ⇒ (p – q) ∈ Z, i.e. (p – q) is an integer.

               ⇒ -(p – q) is an integer

               ⇒ (q – p) is an integer

               ⇒ (q, p) ∈ R

Thus, (p, q) ∈ R ⇒ (q, p) ∈ R

Therefore, R is symmetric.

4. Let m be given fixed positive integer.

Let R = {(p, q) : p, q  ∈ Z and (p – q) is divisible by m}.

Show that R is symmetric relation.

Solution:

Given R = {(p, q) : p, q ∈ Z, and (p – q) is divisible by m}.

Let (p, q) ∈ R . Then,

                ⇒ (p – q) is divisible by m

               ⇒ -(p – q) is divisible by m

               ⇒ (q – p) is divisible by m

               ⇒ (q, p) ∈ R

Thus, (p, q) ∈ R ⇒ (q, p) ∈ R

Therefore, R is symmetric relation on set Z.