N is the set of positive integers and - be a relation on N- N defined a-b - c-d iff ad-bc-Check the relation for being an equivalence relation-

The correct option is A True ( a,b )∼ ( c,d ) iff ad=bc Reflexivity:We can write ab=ab ⇒ (a,b) ∼ (a,b) nbsp; nbsp;(by given def)He ...

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Standard XII

Mathematics

Transitive Relations

N is the set ...

Question

N is the set of positive integers and $\sim$ be a relation on $N \times N d e f i n e d (a, b) \sim (c, d)$ iff ad=bc.
Check the relation for being an equivalence relation.

True

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False

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Solution

The correct option is A True
$(a, b) \sim (c, d)$ iff $a d = b c$
Reflexivity:
We can write $a b = a b$
$\Rightarrow (a, b) \sim (a, b)$ (by given def)
Hence, $\sim$ is reflexive.
Symmetry:
Let $(a, b) \sim (c, d)$
$\Rightarrow a d = b c$
$\Rightarrow d a = c b$ (Product of real numbers is commutative)
$o r c b = d a$
$\Rightarrow (c, d) \sim (a, b)$
Hence, $\sim$ is symmetric.
Transitivity:
Let $(a, b) \sim (c, d); (c, d) \sim (e, f)$
$\Rightarrow a d = b c; c f = d e$
$\Rightarrow a f = b e$ (Since $\frac{d}{c} = \frac{f}{e}$ )
$\Rightarrow (a, b) \sim (e, f)$
Hence, $\sim$ is transitive.
Hence, $\sim$ is an equivalence relation.

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N is the set of positive integers and ∼ be a relation on N×Ndefined(a,b)∼(c,d) iff ad=bc.Check the relation for being an equivalence relation.

N is the set of positive integers and $\sim$ be a relation on $N \times N d e f i n e d (a, b) \sim (c, d)$ iff ad=bc.
Check the relation for being an equivalence relation.