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
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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