Question
Download Solution PDFLet A be {I, m, n}. Let the relation R be {}. Which of the following statements about R is true?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Reflexive: A relation is said to be reflexive if (a, a) ∈ R, for every a ∈ A.
Ex. The relation R in the set {1, 2, 3} given by
R = {(1,1), (2,2), (3,3)} is reflexive.
Symmetric: A relation is said to be symmetric, if (a, b) ∈ R, then (b, a) ∈ R.
Ex. The relation R in the set {a, b, c} given by
R = {(a,b), (b,a), (b,c), (c,b)} is symmetric.
Transitive: A relation is said to be transitive if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
Ex. The relation R in the set {1, 2, 3} given by
R = {(1,2), (2,3), (1,3)}
Explanation:
Given that, A = {l, m, n} and the relation R be {}.
This relation is called a void relation or empty relation on A.
In other words, a relation R on set A is called an empty relation, if no element of A is related to any other element of A.
So, R will be the empty set.
And, R will be the void relation on set A.
So, void relation is not reflexive because it does not contain (a, a) for any a ∈ A.
As we know the definition of symmetric relation is that if A is a set in which the relation R is defined.
Then R is said to be a symmetric relation if (a, b) ∈ R ⇒ (b, a) ∈ R.
Now for void relation R does not contain any element of set A. So, relation R will be trivially symmetric.
As we know the definition of transitive relation is that a relation R over a set A is transitive if for all elements in A.
Whenever R relates a to b and b to c, then R also relates a to c.
So, a void relation has no element. So, it will also be trivially transitive.
So, void relation (or empty relation) is not reflexive but is symmetric and transitive.
Last updated on May 26, 2025
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