Question

A relation $R$ on the set $A = \{1, 2, 3\}$ defined as $R = \{(1, 2), (2, 1), (2, 2)\}$ is:

• A. Reflexive only
• B. Reflexive and Transitive
• C. Symmetric and Transitive
• D. Symmetric only

Hint:

A single counterexample like $(1,2) \text{ and } (2,1) \Rightarrow (1,1) \notin R$ is enough to completely disprove transitivity!

Answer 1

The Correct Option is D

Concept: Let us review the mathematical definitions for types of relations on a set $A$:
• Reflexive: For all $a \in A$, $(a, a) \in R$.
• Symmetric: If $(a, b) \in R$, then $(b, a) \in R$.
• Transitive: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

Step 1: Check for Reflexivity.
The set is $A = \{1, 2, 3\}$. For $R$ to be reflexive, it must contain elements $(1,1), (2,2),$ and $(3,3)$. Looking at $R = \{(1, 2), (2, 1), (2, 2)\}$, we see that $(1,1) \notin R$ and $(3,3) \notin R$. Therefore, $R$ is not reflexive.

Step 2: Check for Symmetry.
Let us check every element pair in $R$:
• For $(1,2) \in R$, its flipped pair is $(2,1)$, which is also in $R$.
• For $(2,1) \in R$, its flipped pair is $(1,2)$, which is also in $R$.
• For $(2,2) \in R$, its flipped pair is $(2,2)$, which is also in $R$. Since every pair has its reverse pair present in the relation, $R$ is symmetric.

Step 3: Check for Transitivity.
For transitivity, if $(a,b) \in R$ and $(b,c) \in R$, then $(a,c)$ must be in $R$. Let us take pairs $(1,2) \in R$ and $(2,1) \in R$. Here, $a=1, b=2, c=1$. For the relation to be transitive, the pair $(a,c) = (1,1)$ must belong to $R$. However, looking closely at the set, $(1,1) \notin R$. Therefore, $R$ is not transitive. Conclusively, the relation is symmetric only, which matches option (D).