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Show that the relation R on $\mathbb {R}$ defined as $R = \{(a,b) : a \leq b\}$ , is reflexive, and transitive but not symmetric.

Answers (1)

$R = \{(a,b) : a \leq b\}$ where R is real numbers

Reflexive:

$\\(a,a)\in R,\ \forall a\in R \\ \therefore a \leq a$

It is reflexive.

Symmetric:

$(a,b)\in R$ then $(b,a)\in R, \ \forall a,b\in R$

Let us assume $a = 2 \qquad b = 3$

$(2,3) \in R \Rightarrow 2 \leq 3 \quad [a\leq b]$

$\Rightarrow (2,3) \in R = LHS$

$(3,2) \Rightarrow 3\nleq 2 \qquad [b\leq a] = RHS$

$\because LHS \neq RHS$

It is not symmetric

Transitive: $(a,b)\in R, \quad (b,c)\in R \Rightarrow (a,c)\in R$

$a\leq b \quad -(1) \qquad b\leq c \quad -(2) \Rightarrow a\leq c \quad -(3)$

From (1) and (2), if $a\leq b$ & $b\leq c$ $\Rightarrow a \leq b \leq c$

$\therefore a\leq c$

Satisfying condition (3)

It is transitive.

Posted by

Ravindra Pindel

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