Question

The relation R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)} on the set A={1,2,3} is
A. reflexive but not symmetric
B. reflexive but not transitive
C. symmetric and transitive
D. neither symmetric nor transitive

Answer 1

Hint- In this question, we use the basic theory of relation. By using some predefined definition, we analyze this relation. A relation R in a set A is called-
Reflexive- if (a,a) ∈ R, for every a ∈ A.
Symmetric- if ($a_1$,$a_2$) ∈ R implies that ($a_2$,$a_1$) ∈ R , for all $a_1$,$a_2$∈ A.
Transitive- if ($a_1$,$a_2$) ∈ R and ($a_2$,$a_3$) ∈ R implies that ($a_1$,$a_3$) ∈ R for all $a_1$,$a_2$,$a_3$ ∈ A.
Equivalence Relation- A relation in a set A is equivalence relation if R is reflexive, symmetric and transitive.

Complete step-by-step answer:
Now,
As given in question-
relation R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}
We have A={1,2,3} and R={(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)}.
Since (1,1),(2,2),(3,3)∈R, R is reflexive
R is not symmetric because (1,2)∈R is true but (2,1)​∈R is not true.
R is transitive because (1,2),(2,3)∈R and (1,3)∈R.
∴ The correct answer is A.

Note- always remember If each element of A is related to every element of A, i.e. R = A × A, then the relation is said to be a universal relation. And also, if no element of A is related to any element of A, i.e. R = φ ⊂ A × A, then the relation in a set is called empty relation.