Question

A relation $R$ on the set $\{1, 2, 3\}$ defined as $R = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)\}$ is given. Two statements are provided: an Assertion (A) and a Reason (R). Select the correct answer from the codes (A), (B), (C), and (D) given below: Assertion (A): A relation $R$ on the set $\{1, 2, 3\}$ defined as $R = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)\}$ is an equivalence relation.
Reason (R): A relation that is reflexive, symmetric and transitive is an equivalence relation.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

To test equivalence relations efficiently on a small finite set: - First, check if all identity components $(x,x)$ are fully present for the set elements. - Second, ignore identity pairs while checking for symmetry and transitivity to save computation time, as identity pairs $(x,x)$ always satisfy symmetry and transitivity automatically.

Answer 1

The Correct Option is A

Concept: An equivalence relation is a fundamental concept in set theory that groups elements sharing a common property. For a relation $R$ defined on a non-empty set $A$ to qualify as an equivalence relation, it must simultaneously satisfy three independent foundational mathematical criteria:
• Reflexivity: Every element in the set must be related to itself. Mathematically, $\forall x \in A, \, (x, x) \in R$.
• Symmetry: If an element is related to another element, then the second element must also be related back to the first. Mathematically, if $(x, y) \in R$, then $(y, x) \in R$ for all $x, y \in A$.
• Transitivity: If a first element relates to a second, and that second element relates to a third, then the first element must relate directly to the third. Mathematically, if $(x, y) \in R$ and $(y, z) \in R$, then $(x, z) \in R$ for all $x, y, z \in A$.

Step 1: Analysis of the fundamental definition given in Reason (R).
The statement given in Reason (R) states: "A relation that is reflexive, symmetric and transitive is an equivalence relation." This statement is directly extracted from the core axiomatic definitions of set theory. By definition, a relation earns the designation of an "equivalence relation" if and only if it possesses all three properties: reflexivity, symmetry, and transitivity. Thus, the statement in Reason (R) is undeniably True.

Step 2: Verifying the Reflexive property for Assertion (A).
The given underlying set is $A = \{1, 2, 3\}$. For the relation $R$ to be reflexive on set $A$, every individual element belonging to $A$ must map to itself within the relation. This means the ordered pairs $(1, 1)$, $(2, 2)$, and $(3, 3)$ must all be explicitly present as elements inside the set $R$. Let us verify the elements of the given relation: \[ R = \{(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)\} \] Checking the elements one by one:
• For $1 \in A$, we observe that $(1, 1) \in R$.
• For $2 \in A$, we observe that $(2, 2) \in R$.
• For $3 \in A$, we observe that $(3, 3) \in R$. Since all required reflexive pairs exist within $R$, the relation is successfully verified to be Reflexive.

Step 3: Verifying the Symmetric property for Assertion (A).
For the relation $R$ to be symmetric, for any ordered pair $(x, y)$ that exists inside $R$, its inverse pair $(y, x)$ must also exist inside $R$. Let us systematically test every distinct non-diagonal ordered pair present in $R$:
• Consider the ordered pair $(1, 2)$. We check if its reverse pair $(2, 1)$ exists. Looking at $R$, we see that $(2, 1) \in R$.
• Consider the ordered pair $(2, 1)$. We check if its reverse pair $(1, 2)$ exists. Looking at $R$, we see that $(1, 2) \in R$.
• For the diagonal elements $(1, 1)$, $(2, 2)$, and $(3, 3)$, their respective reversals are identical to themselves, which naturally belong to $R$. Since every element satisfies the condition of having its corresponding symmetric pair present, the relation is verified to be Symmetric.

Step 4: Verifying the Transitive property for Assertion (A).
For the relation $R$ to be transitive, whenever two pairs of the form $(x, y)$ and $(y, z)$ belong to $R$, the resulting terminal pair $(x, z)$ must also belong to $R$. Let us scrutinize the non-trivial combinations where the second coordinate of the first pair matches the first coordinate of the second pair:
• Combination 1: Take $(1, 2) \in R$ and $(2, 1) \in R$. Here, $x=1, y=2, z=1$. The required transitivity pair is $(x, z) = (1, 1)$. Checking the relation, we find that $(1, 1) \in R$.
• Combination 2: Take $(2, 1) \in R$ and $(1, 2) \in R$. Here, $x=2, y=1, z=2$. The required transitivity pair is $(x, z) = (2, 2)$. Checking the relation, we find that $(2, 2) \in R$.
• Checking with diagonal components like $(1,1)$ combined with $(1,2)$ yields $(1,2)$, which is trivially in $R$. Similarly, $(2,1)$ combined with $(1,1)$ yields $(2,1)$, which is in $R$. Since all possible chain combinations yield valid ordered pairs within $R$, the relation is verified to be Transitive.

Step 5: Synthesizing the final logic conclusion.
Because the relation $R$ has been proven to be completely Reflexive, Symmetric, and Transitive, it satisfies all criteria needed to be classified as an equivalence relation. Therefore, Assertion (A) is True. Furthermore, the reason why Assertion (A) is true is precisely because it fulfills the core definition outlined in Reason (R). Hence, Reason (R) is the exact and correct explanation for Assertion (A).