Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Solution and Explanation
Let A = {1, 2, 3}.
A relation R on A is defined as R = {(1, 2), (2, 1)}.
It is seen that (1, 1), (2, 2), (3, 3) ∉R.
∴ R is not reflexive.
Now, as (1, 2) ∈ R and (2, 1) ∈ R, then R is symmetric.
Now, (1, 2) and (2, 1) ∈ R
However, (1, 1) ∉ R
∴ R is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.
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Concepts Used:
Types of Relation
TYPES OF RELATION
Empty Relation
Relation is said to be empty relation if no element of set X is related or mapped to any element of X i.e, R = Φ.
Universal Relation
A relation R in a set, say A is a universal relation if each element of A is related to every element of A.
R = A × A.
Identity Relation
Every element of set A is related to itself only then the relation is identity relation.
Inverse Relation
Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1
Reflexive Relation
If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.
Symmetric Relation
A relation R is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.
Transitive Relation
A relation is said to be transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A
Equivalence Relation
A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.