let w denote the words in the english dictionary define the

Let $$W$$ denote the words in the English dictionary. Define the relation $$R$$ by $$R = \left\{ {\left( {x,y} \right)} \right. \in W \times W$$ the words $$x$$ and $$y$$ have at least one letter in common.} Then $$R$$ is

A. not reflexive, symmetric and transitive

B. reflexive, symmetric and not transitive

C. reflexive, symmetric and transitive

D. reflexive, not symmetric and transitive

Answer : reflexive, symmetric and not transitive

Solution :
Clearly $$(x, x)$$ $$ \in R\,\forall x \in W.$$ So $$R$$ is relexive.
Let $$(x, y)$$ $$ \in R,$$ then $$(y, x)$$ $$ \in R$$ as $$x$$ and $$y$$ have at least one letter in common. So, $$R$$ is symmetric.
But $$R$$ is not transitive for example
Let $$x$$ = INDIA, $$y$$ = BOMBAY and $$z$$ = JOKER
then $$(x, y)$$ $$ \in R$$ ($$A$$ is common) and $$(y, z)$$ $$ \in R$$ ($$O$$ is common) but $$(x, z)$$ $$ \notin R.$$ (as no letter is common)

Releted MCQ Question on
Calculus >> Sets and Relations

Releted Question 1

If $$X$$ and $$Y$$ are two sets, then $$X \cap {\left( {X \cup Y} \right)^c}$$ equals.

A. $$X$$

B. $$Y$$

C. $$\phi $$

D. None of these

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Releted Question 2

The expression $$\frac{{12}}{{3 + \sqrt 5 + 2\sqrt 2 }}$$ is equal to

A. $$1 - \sqrt 5 + \sqrt 2 + \sqrt {10} $$

B. $$1 + \sqrt 5 + \sqrt 2 - \sqrt {10} $$

C. $$1 + \sqrt 5 - \sqrt 2 + \sqrt {10} $$

D. $$1 - \sqrt 5 - \sqrt 2 + \sqrt {10} $$

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Releted Question 3

If $${x_1},{x_2},.....,{x_n}$$ are any real numbers and $$n$$ is any positive integer, then

A. $$n\sum\limits_{i = 1}^n {{x_i}^2 < {{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$

B. $$\sum\limits_{i = 1}^n {{x_i}^2 \geqslant {{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$

C. $$\sum\limits_{i = 1}^n {{x_i}^2 \geqslant n{{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^2}} $$

D. none of these

View Answer

Releted Question 4

Let $$S$$ = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of $$S$$ is equal to

A. 25

B. 34

C. 42

D. 41

View Answer

Let $$W$$ denote the words in the English dictionary. Define the relation $$R$$ by $$R = \left\{ {\left( {x,y} \right)} \right. \in W \times W$$ the words $$x$$ and $$y$$ have at least one letter in common.} Then $$R$$ is

Releted MCQ Question on
Calculus >> Sets and Relations

If $$X$$ and $$Y$$ are two sets, then $$X \cap {\left( {X \cup Y} \right)^c}$$ equals.

The expression $$\frac{{12}}{{3 + \sqrt 5 + 2\sqrt 2 }}$$ is equal to

If $${x_1},{x_2},.....,{x_n}$$ are any real numbers and $$n$$ is any positive integer, then

Let $$S$$ = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of $$S$$ is equal to

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Let $$W$$ denote the words in the English dictionary. Define the relation $$R$$ by $$R = \left\{ {\left( {x,y} \right)} \right. \in W \times W$$ the words $$x$$ and $$y$$ have at least one letter in common.} Then $$R$$ is

Releted MCQ Question on Calculus >> Sets and Relations

If $$X$$ and $$Y$$ are two sets, then $$X \cap {\left( {X \cup Y} \right)^c}$$ equals.

The expression $$\frac{{12}}{{3 + \sqrt 5 + 2\sqrt 2 }}$$ is equal to

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