Class JEE Mathematics Sets, Relations, and Functions Q #1012
KNOWLEDGE BASED
APPLY
4 Marks 2025 JEE Main 2025 (Online) 23rd January Morning Shift MCQ SINGLE
Let $R = \{(1, 2), (2, 3), (3, 3)\}$ be a relation defined on the set $\{1, 2, 3, 4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, is:
(A) 9
(B) 8
(C) 7
(D) 10
Prev Next
Correct Answer: C
Explanation
For $R$ to be an equivalence relation on $A = \{1, 2, 3, 4\}$, it must be reflexive, symmetric and transitive.

1. **Reflexive:** $R$ must contain $(1, 1), (2, 2), (3, 3), (4, 4)$. Since $(3,3)$ is already in $R$, we need to add $(1, 1), (2, 2), (4, 4)$.

2. **Symmetric:** $R$ must contain $(2, 1)$ and $(3, 2)$ because it contains $(1, 2)$ and $(2, 3)$.

3. **Transitive:** Since $(1, 2)$ and $(2, 3)$ are in $R$, $(1, 3)$ must also be in $R$. And since we added $(3,2)$ now we must add $(1,2)$. Which already exists.

So, the minimum elements to be added are:
$(1, 1), (2, 2), (4, 4), (2, 1), (3, 2), (1, 3)$.

Therefore, the minimum number of elements to be added is $7$.

More from this Chapter

MCQ_SINGLE
Consider the relations $R_1$ and $R_2$ defined as $aR_1b \Leftrightarrow a^2 + b^2 = 1$ for all $a, b \in R$ and $(a, b)R_2(c, d) \Leftrightarrow a+ d = b + c$ for all $(a, b), (c, d) \in N \times N$. Then:
NUMERICAL
Let $A=\{1,2,3, \ldots, 20\}$. Let $R_1$ and $R_2$ two relation on $A$ such that $R_1=\{(a, b): b$ is divisible by $a\}$ $R_2=\{(a, b): a$ is an integral multiple of $b\}$. Then, number of elements in $R_1-R_2$ is equal to _____________.
MCQ_SINGLE
Let $P(S)$ denote the power set of $S=${$1, 2, 3, …, 10$}. Define the relations $R_1$ and $R_2$ on $P(S)$ as $AR_1B$ if $(A \cap B^c) \cup (B \cap A^c) = \emptyset$ and $AR_2B$ if $A \cup B^c = B \cup A^c$, $\forall A, B \in P(S)$. Then :
NUMERICAL
Let $A=\{1,2,3, \ldots \ldots \ldots \ldots, 100\}$. Let $R$ be a relation on $\mathrm{A}$ defined by $(x, y) \in R$ if and only if $2 x=3 y$. Let $R_1$ be a symmetric relation on $A$ such that $R \subset R_1$ and the number of elements in $R_1$ is $\mathrm{n}$. Then, the minimum value of $\mathrm{n}$ is _________.
NUMERICAL
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ___________.
View All Questions