Question

Let $X = \{a_1, a_2, a_3, \ldots, a_n\}$ be a set consisting of $n$ elements. The relation $R = \{(a_1,a_1),(a_2,a_2),(a_3,a_3),\ldots,(a_n,a_n)\}$ on the set $X$ is:

• A. reflexive, symmetric but not transitive
• B. reflexive, transitive but not symmetric
• C. transitive, symmetric but not reflexive
• D. reflexive, symmetric and transitive
• E. reflexive, not symmetric and not transitive

Hint:

A relation containing only identity pairs $(a,a)$ is always reflexive, symmetric, and transitive.

Answer 1

The Correct Option is D

Concept: A relation $R$ on a set $X$ is:
• Reflexive: if $(a,a) \in R$ for all $a \in X$
• Symmetric: if $(a,b) \in R \Rightarrow (b,a) \in R$
• Transitive: if $(a,b),(b,c) \in R \Rightarrow (a,c) \in R$

Step 1: Check Reflexivity
All elements $(a_i,a_i)$ are present in $R$.
Hence, relation is reflexive.

Step 2: Check Symmetry
Each ordered pair is of the form $(a_i,a_i)$, so its reverse $(a_i,a_i)$ is also present.
Hence, relation is symmetric.

Step 3: Check Transitivity
If $(a_i,a_i)$ and $(a_i,a_i)$ are in $R$, then $(a_i,a_i)$ is also in $R$.
Hence, relation is transitive. Final Conclusion:
The relation is reflexive, symmetric and transitive.