Here you will learn what is symmetric relation on sets with definition and examples.
Let’s begin –
What is Symmetric Relation ?
Definition : A relation R on a set A is said to be a symmetric relation iff
(a, b) ∈ R ⟹ (b, a) ∈ R for all a, b ∈ A
i.e. a R b ⟹ b R a for all a, b ∈ A.
Note :
(i) The identity and the universal relations on a non-void set are symmetric relations.
(ii) A relation R on the set A is not a symmetric relation if there are at least two elements a, b ∈ A such that (a, b) ∈ R but (b, a) ∈ R.
Also Read : Types of Relations in Math
Given below are some symmetric relation examples.
Example : Let A = {1, 2, 3, 4} and let R1 and R2 be relations on A given by R1 = {(1, 3),(1, 4),(3, 1),(2, 2),(4, 1)} and R2 = {(1, 1),(2, 2),(3, 3),(1, 3)}. Clearly, R1 is a symmetric relation on A. However, R2 is not so, because (1, 3) ∈ R2 but (3, 1) ∉ R2.
Example : Let S be a non-void set and R be a relation defined on power set P(S) by (A, B) ∈ R ⟺ A ⊆ B for all A, B ∈ P(S). Then, R is not a symmetric relation.
Note : A reflexive relation on a set A is not necessarily symmetric. For example, the relation R = {(1, 1),(2, 2),(3, 3),(1, 3)} is a reflexive relation on set A = {1, 2, 3} but it is not symmetric.