Question
Download Solution PDFSuppose X = {1, 2, 3, 4} and R is a relation on X.
If R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}, then which one of the following is correct?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
For any two non-empty sets A and B, a relation R from A to B is a subset of the cartesian product A X B.
Reflexive:
Let R be a relation on a non-empty set A, if every element of A is related to itself then R is said to be a reflexive relation.
Thus, R is reflexive if and only if (a, a) ∈ R, ∀ a ∈ A.
Symmetric:
Let R be a relation on a non-empty set A, then the relation R is said to be symmetric relation if and only if (a, b) ∈ R ⇒ (b, a) ∈ R
Transitive:
Let R be a relation on a non-empty set A, then the relation R is said to be transitive relation if and only if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R
Equivalence:
Let R be a relation on a non-empty set A, then the relation R is said to be equivalence relation if R is reflexive, symmetric and transitive.
Calculation:
Given: X = {1, 2, 3, 4} and R is a relation on X given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}
Reflexive:
The given relation R on X is not reflexive ∵ (a, a) ∉ R, ∀ a ∈ X i.e (4, 4) ∉ R.
Symmetric:
The given relation R on X is symmetric ∵ for all (a, b) ∈ R ⇒ (b, a) ∈ R
Transitive:
The given relation R on X is transitive ∵ for all (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ RLast updated on May 30, 2025
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