Class JEE Mathematics Sets, Relations, and Functions Q #1075
KNOWLEDGE BASED
APPLY
4 Marks 2006 AIEEE MCQ SINGLE
Let $W$ denote the words in the English dictionary. Define the relation $R$ by $R = {(x, y) ∈ W × W |$ the words $x$ and $y$ have at least one letter in common}. Then, $R$ is
(A) reflexive, symmetric and not transitive
(B) reflexive, symmetric and transitive
(C) reflexive, not symmetric and transitive
(D) not reflexive, symmetric and transitive
Prev Next
Correct Answer: A
Explanation
To determine the properties of the relation $R$, we analyze reflexivity, symmetry, and transitivity.

Reflexivity: A word always shares at least one letter with itself. So, $(x, x) ∈ R$ for all $x ∈ W$. Therefore, $R$ is reflexive.

Symmetry: If a word $x$ has a letter in common with word $y$, then $y$ also has a letter in common with $x$. So, if $(x, y) ∈ R$, then $(y, x) ∈ R$. Therefore, $R$ is symmetric.

Transitivity: Consider the words 'cat', 'bat', and 'bee'. 'cat' and 'bat' share the letter 'a', so (cat, bat) ∈ R. 'bat' and 'bee' share the letter 'b', so (bat, bee) ∈ R. However, 'cat' and 'bee' do not share any common letters, so (cat, bee) ∉ R. Therefore, $R$ is not transitive.

Thus, the relation $R$ is reflexive, symmetric, and not transitive.

More from this Chapter

MCQ_SINGLE
Let $R$ be a relation on $\mathbb{R}$, given by $R = \{(a, b) : 3a - 3b + \sqrt{7} \text{ is an irrational number} \}$. Then $R$ is
NUMERICAL
Let $$A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $$ and $$B = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\max \,\{ i,j\} } } $$Then A + B is equal to _____________.
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 ___________.
NUMERICAL
The number of elements in the set $\left\{n \in \mathbb{N}: 10 \leq n \leq 100\right.$ and $3^{n}-3$ is a multiple of 7$\}$ 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