Question

The relation \( R = \{(1,1),(2,2),(3,3)\ \) on the set \( \{1,2,3\} \) is}

• A. symmetric only
• B. an equivalence relation
• C. transitive only
• D. reflexive only

Hint:

A relation is equivalence if it satisfies all three: reflexive, symmetric, and transitiveAlways check each property step by step.

Answer 1

The Correct Option is B

Step 1: Check reflexive property.
A relation is reflexive if:
\[ (a,a) \in R \text{ for all } a \in \{1,2,3\} \]
Given:
\[ (1,1),(2,2),(3,3) \in R \]
So, \( R \) is reflexive.

Step 2: Check symmetric property.
A relation is symmetric if:
\[ (a,b) \in R \Rightarrow (b,a) \in R \]
Here all pairs are of the form \( (a,a) \), so symmetry holds automatically.

Step 3: Check transitive property.
A relation is transitive if:
\[ (a,b),(b,c) \in R \Rightarrow (a,c) \in R \]
Since all elements are of type \( (a,a) \), transitivity is satisfied.

Step 4: Combine all properties.
Relation is reflexive, symmetric, and transitive.

Step 5: Definition.
Such a relation is called an equivalence relation.

Step 6: Match with options.
Correct option is (B).

Step 7: Final conclusion.
\[ \boxed{\text{an equivalence relation}} \]