Q.The relation R in the set $\{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1), (1, 1)\}$ is (A) symmetric and transitive, but not reflexive (B) reflexive and symmetric, but not transitive (C) symmetric, but neither reflexive nor transitive (D) an equivalence relation

The key idea involves checking the properties of a relation: reflexivity, symmetry, and transitivity.

Given the set $A = \{1, 2, 3\}$ and the relation $R = \{(1, 2), (2, 1), (1, 1)\}$.

1. Reflexivity: A relation $R$ on set $A$ is reflexive if $(a, a) \in R$ for all $a \in A$.

   Here, $(2, 2) \notin R$ and $(3, 3) \notin R$. Thus, $R$ is not reflexive.

2. Symmetry: A relation $R$ is symmetric if whenever $(a, b) \in R$, then $(b, a) \in R$.

   For $(1, 2) \in R$, we have $(2, 1) \in R$. For $(2, 1) \in R$, we have $(1, 2) \in R$. For $(1, 1) \in R$, we have $(1, 1) \in R$. Thus, $R$ is symmetric.

3. Transitivity: A relation $R$ is transitive if whenever $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

   Consider $(2, 1) \in R$ and $(1, 2) \in R$. For transitivity, $(2, 2)$ must be in $R$. However, $(2, 2) \notin R$. Thus, $R$ is not transitive.

The relation $R$ is symmetric, but neither reflexive nor transitive. This matches option (C).

To determine the properties of the given relation $R$ on the set $A = \{1, 2, 3\}$, we need to check if it satisfies the definitions of reflexivity, symmetry, and transitivity. Understanding these definitions precisely is key to avoiding common errors.

A relation $R$ on a set $A$ is:

• Reflexive if for every element $a \in A$, the ordered pair $(a, a)$ is in $R$. This means every element must be related to itself.
• Symmetric if for every pair $(a, b) \in R$, the pair $(b, a)$ is also in $R$. This means if $a$ is related to $b$, then $b$ must also be related to $a$.
• Transitive if for every $a, b, c \in A$, whenever $(a, b) \in R$ and $(b, c) \in R$, it must follow that $(a, c) \in R$. This means if $a$ is related to $b$ and $b$ is related to $c$, then $a$ must also be related to $c$.

Let's examine $R = \{(1, 2), (2, 1), (1, 1)\}$ on the set $A = \{1, 2, 3\}$ step by step.

1. Check for Reflexivity:

   For $R$ to be reflexive, every element in $A$ must be related to itself. That is, $(1, 1)$, $(2, 2)$, and $(3, 3)$ must all be present in $R$.

   • We see that $(1, 1) \in R$.
   • However, $(2, 2) \notin R$.
   • Also, $(3, 3) \notin R$. Since $(2, 2)$ and $(3, 3)$ are not in $R$, the relation $R$ is not reflexive.

2. Check for Symmetry:

   For $R$ to be symmetric, for every pair $(a, b) \in R$, the reverse pair $(b, a)$ must also be in $R$.

   Let's check each pair in $R$:

   • For $(1, 2) \in R$, we need to check if $(2, 1) \in R$. Yes, $(2, 1) \in R$.
   • For $(2, 1) \in R$, we need to check if $(1, 2) \in R$. Yes, $(1, 2) \in R$.
   • For $(1, 1) \in R$, we need to check if $(1, 1) \in R$. Yes, $(1, 1) \in R$. Since for every ordered pair $(a, b)$ in $R$, the pair $(b, a)$ is also in $R$, the relation $R$ is symmetric.

3. Check for Transitivity:

   For $R$ to be transitive, if $(a, b) \in R$ and $(b, c) \in R$, then $(a, c)$ must also be in $R$. We need to check all such combinations.

   • Consider $(1, 2) \in R$ and $(2, 1) \in R$. Here, $a=1, b=2, c=1$. According to the definition of transitivity, $(a, c) = (1, 1)$ must be in $R$. We see that $(1, 1) \in R$. This case holds.
   • Consider $(2, 1) \in R$ and $(1, 2) \in R$. Here, $a=2, b=1, c=2$. According to the definition of transitivity, $(a, c) = (2, 2)$ must be in $R$. However, we found earlier that $(2, 2) \notin R$. Since we found a case where $(2, 1) \in R$ and $(1, 2) \in R$, but $(2, 2) \notin R$, the relation $R$ is not transitive.
   Watch out

   A common mistake when checking transitivity is to only look for pairs that "chain" and forget to check if the resulting pair exists. If even one such chain $(a,b), (b,c)$ exists where $(a,c)$ is missing, the relation is not transitive.

In summary:

• $R$ is not reflexive.
• $R$ is symmetric.
• $R$ is not transitive.

Now, let's compare these findings with the given options:

(A) symmetric and transitive, but not reflexive (Incorrect, not transitive)

(B) reflexive and symmetric, but not transitive (Incorrect, not reflexive)

(C) symmetric, but neither reflexive nor transitive (Correct)

(D) an equivalence relation (Incorrect, an equivalence relation must be reflexive, symmetric, and transitive)