Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

Let $S$ be the set of all real numbers and let $R$ be a relation on $S$ defined by $a R b \Leftrightarrow|a| \leq b$. Then, $R$ is

A. reflexive but neither symmetric nor transitive

B. symmetric but neither reflexive nor transitive

C. transitive but neither reflexive nor symmetric

D. none of these

 

Solution:

According to the question,

Given set $S=\{\ldots \ldots,-2,-1,0,1,2 \ldots \ldots\}$

And $R=\{(a, b): a, b \in S$ and $|a| \leq b\}$

Formula

For a relation $R$ in set $A$

Reflexive

The relation is reflexive if $(a, a) \in R$ for every $a \in A$

Symmetric

The relation is Symmetric if $(a, b) \in R$, then $(b, a) \in R$

Transitive

Relation is Transitive if $(a, b) \in R \&(b, c) \in R$, then $(a, c) \in R$

Equivalence

If the relation is reflexive, symmetric and transitive, it is an equivalence relation.

Check for reflexive

Consider, (a,a)

$\therefore|\mathrm{a}| \leq \mathrm{a}$ and which is not always true.

Therefore , R is not reflexive ……. (1)

Check for symmetric

$a R b \Rightarrow|a| \leq b$

$b R a \Rightarrow|b| \leq a$

Both cannot be true.

$E x_{-}$If $a=-2$ and $b=-1$

$\therefore 2 \leq-1$ is false and $1 \leq-2$ which is also false.

Therefore , R is not symmetric ……. (2)

Check for transitive

$a R b \Rightarrow|a| \leq b$

$b R c \Rightarrow|b| \leq c$

$\therefore|\mathrm{a}| \leq \mathrm{c}$

Ex $_{-} \mathrm{a}=-5, \mathrm{~b}=7$ and $\mathrm{c}=9$

$\therefore 5 \leq 7,7 \leq 9$ and hence $5 \leq 9$

Therefore , R is transitive ……. (3)

Now, according to the equations $(1),(2),(3)$

Correct option will be (C)

 

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