Question

Let $R = \{(1,1),(2,2),(3,3),(1,2)\}$ be a relation on $\{1,2,3\}$. The minimum number of elements to be added so that $R$ is an equivalence relation is:}

• A. 4
• B. 3
• C. 5
• D. 1

Hint:

For equivalence relations, symmetry is usually the key missing condition.

Answer 1

The Correct Option is A

Concept: An equivalence relation must be reflexive, symmetric, and transitive.

Step 1: {Check reflexive property.}
Required pairs: \[ (1,1),(2,2),(3,3) \] All present ✔

Step 2: {Check symmetry.}
Given $(1,2)$ exists, so $(2,1)$ must also exist. Add: \[ (2,1) \]

Step 3: {Check transitivity.}
Since $(2,1)$ and $(1,2)$ exist, $(2,2)$ already exists ✔ No further required pairs for closure.

Step 4: {Count additions.}
Only one element is needed: \[ (2,1) \]