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Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

Let $Z$ be the set of all integers and let $R$ be a relation on $Z$ defined by $a R b \Leftrightarrow a \geq b$. Then, $R$ is

A. symmetric and transitive but not reflexive

B. reflexive and symmetric but not transitive

C. reflexive and transitive but not symmetric

D. an equivalence relation

 

Solution:

According to the question ,

Given set $Z=\{1,2,3,4 \ldots . .\}$

And $R=\{(a, b): a, b \in Z$ and $a \geq b\}$

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, $(a, a)(b, b)$

$\therefore \mathrm{a} \geq \mathrm{a}$ and $\mathrm{b} \geq \mathrm{b}$ which is always true.

Therefore , R is reflexive ……. (1)

Check for symmetric

$a R b \Rightarrow a \geq b$

$b R a \Rightarrow b \geq a$

Both cannot be true.

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

$\therefore 2 \geq 1$ is true but $1 \geq 2$ which is false.

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

Check for transitive

$a R b \Rightarrow a \geq b$

$b R c \Rightarrow b \geq c$

$\therefore a \geq c$

Ex $a=5, b=4$ and $c=2$

$\therefore 5 \geq 4,4 \geq 2$ and hence $5 \geq 2$

Therefore , R is transitive ……. (3)

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

Correct option will be (C)

 

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