Let A = { 2,3,4,5,...,30} and textquotesingle{} simeq ... | Relation (Mathematics)

Exam ( $16^{th\ }$ March $2^{nd\ }$ Shift 2021)

Mathematics

Let

A = \{ 2,3,4,5,\ldots,30\}

and

\simeq

be an equivalence relation on

A \times A

, defined by

(a,b) \simeq (c,d)

if and only if

ad = bc

. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair

(4,3)

is equal to

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Correct Answer:

Explanation

(c) : $(a,b) \simeq (c,d) \Leftrightarrow ad = bc\\$ $\Rightarrow \frac{a}{c} = \frac{b}{d} \Rightarrow \frac{4}{c} \\= \frac{3}{d} \Rightarrow \frac{c}{d} \\= \frac{4}{3}(c,d) \in (2,3,4,\ldots,30)\\$ {23} $\therefore (c,d) \\= (4,3),(8,6),(12,9),(16,12),(20,15),(24,18)$ , $(28,21)\\$ $\therefore 7$ ordered pair exist.

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