Let R be the relation in the set {1,2,3,4} given by R= {(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2)}. Choose the correct answer.
- R is reflexive and symmetric but not transitive.
R is reflexive and transitive but not symmetric.
R is symmetric and transitive but not reflexive.
R is an equivalence relation.
The Correct Option is B
Solution and Explanation
R =\({(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}\)
It is seen that \((a, a) ∈ R\), for every \(a ∈{1, 2, 3, 4}\).
∴ R is reflexive. It is seen that \((1, 2) ∈\)R, but \((2, 1) ∉ R.\)
∴R is not symmetric. Also, it is observed that \((a, b), (b, c) ∈ R ⇒ (a, c) ∈ R\) for all \(a, b, c ∈ {1, 2, 3, 4}\).
∴ R is transitive. Hence, R is reflexive and transitive but not symmetric.
The correct answer is B.
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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.