Let R_1 and R_2 be two relations defined on [R] by [a ,R_1 ,b a b 0] and [a ,R_2 ,b a b] T…

Question

Let $R_{1}$ and $R_{2}$ be two relations defined on $$\mathbb{R}$$ by

$$a \,R_{1} \,b \Leftrightarrow a b \geq 0$$ and $$a \,R_{2} \,b \Leftrightarrow a \geq b$$

Then,

Accepted Answer

$$a\,{R_1}\,b \Leftrightarrow ab \ge 0$$

So, definitely $$(a,a) \in {R_1}$$ as $${a^2} \ge 0$$

If $$(a,b) \in {R_1} \Rightarrow (b,a) \in {R_1}$$

But if $$(a,b) \in {R_1},(b,c) \in {R_1}$$

$$\Rightarrow$$ Then $(a,c)$ may or may not belong to R1

Consider $a =  - 5,b = 0,c = 5$ so $(a,b)$ and $$(b,c) \in {R_1}$$ but $ac 1 is not equivalence relation$a\,{R\_2}\,b \Leftrightarrow a \ge b$$

(a,a) \in {R\_2} \Rightarrow \[ so reflexive relation

If (a,b) \in {R\_2}\[ then (b,a) ( may or may not belong to R2

$$

Solutions · 1