Class JEE Mathematics Sets, Relations, and Functions Q #1010
KNOWLEDGE BASED
APPLY
4 Marks 2025 JEE Main 2025 (Online) 23rd January Evening Shift MCQ SINGLE
Let $X = R \times R$. Define a relation R on X as: $(a_1, b_1) R (a_2, b_2) \Leftrightarrow b_1 = b_2$ Statement I: $R$ is an equivalence relation. Statement II: For some $(a, b) \in X$, the set $S = \{(x, y) \in X : (x, y)R(a, b)\}$ represents a line parallel to $y = x$. In the light of the above statements, choose the correct answer from the options given below:
(A) Both Statement I and Statement II are true
(B) Statement I is true but Statement II is false
(C) Both Statement I and Statement II are false
(D) Statement I is false but Statement II is true
Prev Next
Correct Answer: B
Explanation
Statement I: Reflexive: $(a_1, b_1) R (a_1, b_1) \Rightarrow b_1 = b_1$ (True). Symmetric: $(a_1, b_1) R (a_2, b_2) \Rightarrow b_1 = b_2$ and $(a_2, b_2) R (a_1, b_1) \Rightarrow b_2 = b_1$ (True). Transitive: $(a_1, b_1) R (a_2, b_2) \Rightarrow b_1 = b_2$ and $(a_2, b_2) R (a_3, b_3) \Rightarrow b_2 = b_3$. Thus, $b_1 = b_3$, so $(a_1, b_1) R (a_3, b_3)$ (True). Hence, relation $R$ is an equivalence relation, and Statement I is true. For Statement II: $(x, y) R (a, b) \Rightarrow y = b$. This represents a line $y=b$, which is parallel to the x-axis. It is not parallel to the line $y=x$. Therefore, Statement II is false.

More from this Chapter

MCQ_SINGLE
Let $A = \{-3, -2, -1, 0, 1, 2, 3\}$ and R be a relation on A defined by $xRy$ if and only if $2x - y \in \{0, 1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l + m + n$ is equal to:
NUMERICAL
Let A = {n $ \in $ N: n is a 3-digit number} B = {9k + 2: k $ \in $ N} and C = {9k + $l$: k $ \in $ N} for some $l ( 0 < l < 9)$ If the sum of all the elements of the set A $ \cap $ (B $ \cup $ C) is 274 $ \times $ 400, then $l$ is equal to ________.
MCQ_SINGLE
Let a set $A = A_1 \cup A_2 \cup ..... \cup A_k$, where $A_i \cap A_j = \phi$ for $i \neq j$, $1 \le j, j \le k$. Define the relation R from A to A by $R = \{(x, y) : y \in A_i$ if and only if $x \in A_i, 1 \le i \le k\}$. Then, R 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 ___________.
MCQ_SINGLE
Let $A = {n \in [100, 700] \cap N : n$ is neither a multiple of 3 nor a multiple of 4}. Then the number of elements in $A$ is
View All Questions