Let A be {I, m, n}. Let the relation R be {}. Which of the following statements about R is true?

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  1. R is not reflexive, is symmetric, and is transitive.
  2. R is not reflexive, is not symmetric, and is not transitive.
  3. R is not reflexive, is symmetric, and is not transitive.
  4. R is reflexive, is symmetric, and is not transitive.

Answer (Detailed Solution Below)

Option 1 : R is not reflexive, is symmetric, and is transitive.
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Concept:

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.

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