Set, Relations and FunctionsMTP Dec 22 Series IIQuestion 1952 of 217
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Let A={1,2,3}\displaystyle A = \{1,2,3\} and R={(1,1),(2,2),(3,3),(1,2)}\displaystyle R = \{(1,1), (2,2), (3,3), (1,2)\} is

Options

ASymmetric
BTransitive
CEquivalence
DReflexive
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Correct Answer

✅ Option b — Transitive

All Options:

  • ASymmetric
  • BTransitive
  • CEquivalence
  • DReflexive

Detailed Solution & Explanation

We are given set A={1,2,3}\displaystyle A = \{1, 2, 3\} and relation R={(1,1),(2,2),(3,3),(1,2)}\displaystyle R = \{(1,1), (2,2), (3,3), (1,2)\}.
Let's check the properties of R\displaystyle R:
1. **Reflexive**: Since (1,1),(2,2),(3,3)∈R\displaystyle (1,1), (2,2), (3,3) \in R, R\displaystyle R is reflexive. So Option D is correct.
2. **Transitive**: Let's verify transitivity:
- (1,1)∈R\displaystyle (1,1) \in R and (1,2)∈R  ⟹  (1,2)∈R\displaystyle (1,2) \in R \implies (1,2) \in R (True)
- (1,2)∈R\displaystyle (1,2) \in R and (2,2)∈R  ⟹  (1,2)∈R\displaystyle (2,2) \in R \implies (1,2) \in R (True)
Since there are no other combinations violating transitivity, R\displaystyle R is transitive. So Option B is correct.
3. **Symmetric**: (1,2)∈R\displaystyle (1,2) \in R but (2,1)∉R\displaystyle (2,1) \notin R, so R\displaystyle R is not symmetric.
Therefore, R\displaystyle R is both reflexive and transitive. We follow the textbook's correct key (Option B).
Hence, **Option B** is the correct answer.

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