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Symmetric relation

A symmetric relation is nothing but a type of a binary relation. The relation “is equal to”, is symmetric because if a = b is true then b = a is also true. In formal terms, a binary relation Rover a set X is symmetric only in the following condition:

If RT represents the converse of R, then R is symmetric if and only if R = RT

Let A be a set and R be the relation defined. Then it can be concluded that R is a symmetric relation, if (a, b) ∈ R ⇒ (b, a) ∈ R i.e. aRb equals bRa for all (a, b) ∈ R.

Consider, for example, the set A of natural numbers. Let A be a relation defined by “x + y = 4”, then this relation is symmetric in A, for

a + b = 4 is same as b + a = 4

But in the set A of natural numbers if the relation R be defined as ‘x is a divisor of y’, then the relation R is not symmetric as 3R9 does not imply 9R3; for, 3 divides 9 but 9 does not divide 3.

For a symmetric relation R, R−1−1 = R.

1. A relation R is defined on the set Z by “a R b if a – b is divisible by 5” for a, b ∈ Z. Analyse if R is a symmetric relation on Z.

Solution:

Let a, b ∈ Z and aRb hold. Then a – b is divisible by 5 and therefore b –a is divisible by 5.

Thus, aRb equals bRa and therefore R is symmetric.

2. A relation R is defined on the set Z (set of all integers) by “aRb if and only if 2a + 3b is divisible by 5”, for all a, b ∈ Z. Find out if R is a symmetric relation on Z.

Solution:

Let a, b ∈ Z and aRb holds i.e., 2a + 3a = 5a, which is divisible by 5. Now, 2a + 3a = 5a – 2a + 5b – 3b = 5(a + b) – (2a + 3b) is also divisible by 5.

Hence aRa holds for all a in Z i.e. R is reflexive.

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

Solution:

Given R = {(a, b) : a, b ∈ Q, and a – b ∈ Z}.

Let ab∈ R ⇒ (a – b) ∈ Z, i.e. (a – b) is an integer.

⇒-(a – b) is an integer

⇒ (b – a) is an integer

⇒ (b, a) ∈ R

Thus, (a, b) ∈ R ⇒ (b, a) ∈ R

Therefore, R is symmetric.

4. Let m be given fixed positive integer.

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

Show that R is symmetric relation.

Solution:

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

Let ab∈R . Then,

ab∈ R ⇒ (a – b) is divisible by m

⇒-(a – b) is divisible by m

⇒ (b – a) is divisible by m

⇒ (b, a) ∈ R

Thus, (a, b) ∈ R ⇒ (b, a) ∈ R

Therefore, R is symmetric relation on set Z.

 

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