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JEE Mathematics 2024 Question with Solution

If RR is the smallest equivalence relation on the set {1,2,3,4}\{1, 2, 3, 4\} such that {(1,2),(1,3)}⊆R\{(1, 2), (1, 3)\} \subseteq R, then the number of elements in RR is:

  • A

    1010

  • B

    1212

  • C

    88

  • D

    1515

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The set is {1,2,3,4}\{1,2,3,4\} and {(1,2),(1,3)}⊆R\{(1,2),(1,3)\} \subseteq R.

Find: The number of ordered pairs in the smallest equivalence relation RR.

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

From reflexivity, RR must contain

(1,1),(2,2),(3,3),(4,4)(1,1),(2,2),(3,3),(4,4)

Closure Under Symmetry and Transitivity

Since (1,2)∈R(1,2) \in R and (1,3)∈R(1,3) \in R are given, symmetry gives

(2,1),(3,1)(2,1),(3,1)

Now apply transitivity. Because (2,1)∈R(2,1) \in R and (1,3)∈R(1,3) \in R, we must have

(2,3)∈R(2,3) \in R

By symmetry, this also gives

(3,2)∈R(3,2) \in R

Hence the smallest equivalence relation is

R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)}R = \{(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}

So the total number of elements is 1010. Therefore, the correct option is A.

Common mistakes

  • Adding only the given pairs and reflexive pairs is incorrect, because an equivalence relation must also satisfy symmetry and transitivity. Always complete the closure under all three properties.

  • Forgetting (2,3)(2,3) and (3,2)(3,2) is a common error. Since (2,1)(2,1) and (1,3)(1,3) are in RR, transitivity forces (2,3)(2,3), and then symmetry forces (3,2)(3,2).

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