Question

A void relation in any set A is
A. reflexive
B. reflexive and transitive
C. symmetric and transitive
D. reflexive and symmetric

Answer 1

Hint: Void relation is the same as empty relation. And void relations have no elements. So, it will be the empty relation for set A.

Complete step-by-step answer:
As we know the definition of void relation is that if A be a set, then \[\phi {\text{ }} \subseteq \] A\[ \times \]A and so it is a relation on A. This relation is called 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.
As we are given that the relation is for set A.
So, let us assume any set and any relation R for that set.
Let set A = {1, 2, 3, 4}
And the relation for set A will be R = {(a, b): a + b = 10}
So, we observe that a + b \[ \ne \] 10 for any two elements of set A.
Therefore (a, b) \[ \notin \] R for any a, b \[ \in \] A.
R does not contain elements of A\[ \times \]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\[ \in \]R.
As we know the definition of symmetric relation that if A be a set in which the relation R is defined. Then R is said to be a symmetric relation, if (a, b) \[ \in \] R \[ \Rightarrow \] (b, a) \[ \in \] 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 that a relation R over a set A is transitive if for all elements a, b, c 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 is not reflexive but is symmetric and transitive.
Hence, the correct option will be C.

Note: Whenever we come up with this type of problem then first, we assume any set A and a relation R over set A such that R is a void relation. Then we will go with the definition of reflexive symmetric and transitive relation. But we should note that if R is the empty relation then it will always be symmetric and transitive because it does not contain any element of set A. And relation R is reflexive and transitive if (a, b) \[ \in \] R \[ \Rightarrow \] (b, a) \[ \in \] R and if R relates a to b and b to c, then R also relates a to c. But relation R has none of a, b, and c. So, it will always be true.