CUET PG 2021 — Mathematics PYQ

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Question

CUET PG 2021 — Mathematics PYQ

CUET PG | Mathematics | 2021

Determine the characteristics of the relation aRb, if a² = b²

Choose the correct answer:

Explanation

1. Reflexive (Swayamtulya)

Ek relation reflexive hota hai agar har element $a$ khud se related ho ( $aRa$ ).

Kyunki kisi bhi number ka square hamesha apne aap ke barabar hota hai:

a^2 = a^2

Isliye, $(a, a) \in R$ for all $a$ .

Nishkarsh: Ye relation Reflexive hai.

2. Symmetric (Sammit)

Agar $aRb$ hai, toh kya $bRa$ bhi hoga?

Maana ki $aRb$ hai, toh condition ke hisaab se:

a^2 = b^2

Is equation ko hum aise bhi likh sakte hain:

b^2 = a^2

Iska matlab hai ki $(b, a) \in R$ .

Nishkarsh: Ye relation Symmetric hai.

3. Transitive (Sankramak)

Agar $aRb$ aur $bRc$ hai, toh kya $aRc$ hoga?

Maana:

$a^2 = b^2$ (Kyunki $aRb$ )
$b^2 = c^2$ (Kyunki $bRc$ )

Dono equations se hum keh sakte hain:

a^2 = c^2

Iska matlab hai ki $(a, c) \in R$ .

Nishkarsh: Ye relation Transitive hai.

Final Result:

Kyunki ye relation Reflexive, Symmetric, aur Transitive teeno hai:

\text{The relation } R = \{(a, b) : a^2 = b^2\} \text{ is an \textbf{Equivalence Relation.}}