Question

Which of the below statements is/are correct?

• A. Empty relation is a symmetric relation
• B. "is married to" is an example of symmetric relation
• C. A relation R is defined as ‘x is a divisor of y’
• D. Empty relation is a reflexive relation

Answer 1

Answer: (A), (B)

The correct options are

A

Empty relation is a symmetric relation

B

"is married to" is an example of symmetric relation

We will go through each statement and check if it is a symmetric relation.

1. In an empty relation no element will be related to any element or there won’t be any relation in an empty relation. The condition for a relation to be symmetric is that if there is an element (a,b) there should be an element (b,a). Since there are no elements of the form (a,b) we need not have any element (b,a). So an empty relation will be symmetric.
2. If a is married to b, then we can say that b is married to a also. So the relation will be symmetric because if there is (a,b) , there will be an element (b,a) in the relation.
3. We know that 2 is a divisor of 32, but 32 is not divisor of 2. According to the given relation, we can say that if 2 is related 32, then 32 won't be related 2. Which makes it not symmetric.
4. A relation $R$ is said to be reflexive on set $A$, if $(a,a)\in R$ for every $a\in A$.
   Since, the empty relation contains no elements as it obvious that the elements of the form $(a,a)$ does not available in $R$.
   Hence, empty relation is not a reflexive relation.