First slide

Relations XII

Question

Let R be a relation defined by $R={(a,b):a≥b},$ where a and b are real numbers, then R is

Easy

A

reflexive, symmetric and transitive
B

reflexive, transitive but not symmetric
C

symmetric, transitive but not reflexive
D

neither transitive, nor reflexive, not symmetric

Solution

$R = {(a, b) : a \geq b}$
We know that, $a \geq a$
$∴ (a, a) \in R, \forall a \in R$
R is a reflexive relation.
Let $(a, b) \in R$
$\begin{array}{lrl} \Rightarrow & a \geq b \\ \Rightarrow & b \leq a \\ \Rightarrow & (b, a) \notin R \end{array}$
So, R is not symmetric relation.
Now , let $(a, b) \in R$ and $(b, c) \in R$
$\begin{array}{lc} \Rightarrow & a \geq b and b \geq c \\ \Rightarrow & a \geq c \\ \Rightarrow & (a, c) \in R \end{array}$
R is a transitive relation.

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