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Introduction to Relations

Question

For the relation $$R_1$$ defined on $$R$$ by the rule $$(a, b)\in R_1\Leftrightarrow 1+ab > 0$$. Prove that: $$(a, b)\in R_1$$ and $$(b, c)\in R_1\Rightarrow (a, c)\in R_1$$ is not true for all $$a, b, c \in$$ R.

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Similar Questions

Q1

For the relation

R_{1}

defined on

R

by the rule

(a, b) \in R_{1} \Leftrightarrow 1 + a b > 0

. Prove that :

(a, b) \in R_{1}

and

(b, c) \in R_{1} \Rightarrow (a, c) \in R_{1}

is not true for all

a, b, c \in R

.

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Q2

For the relation $R_{1}$ defined on R by the rule $(a, b) ϵ R_{1} \Leftrightarrow 1 + a b > 0.$

Prove that : $(a, b) ϵ R_{1}$ and $(b, c) ϵ R_{1}$

$\Rightarrow (a, c) ϵ R_{1}$ is not true for all $a, b c ϵ R$

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Q3

Let

R_{1}

and

R_{2}

be two relations defined on

R

by a

R_{1} b \Leftrightarrow a b \geq 0

and a

R_{2} b \Leftrightarrow a \geq b

. Then,

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Q4

The relation r1 is defined on a set A={a,b,c} as follows
R={(a,a)}

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Q5

Consider the following two binary relations on the set

A = {a, b, c} :

R_{1} = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and R_{2} = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)} .

Then :

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