JEE 2023 — Mathematics PYQ

Ek relation ko equivalence relation hone ke liye Reflexive, Symmetric aur Transitive hona zaroori hai. Chaliye ek-ek karke check karte hain:

1. Reflexive:

Kisi bhi $a\in \mathbb{N}$ ke liye, humein check karna hai ki kya $aRa$ sahi hai.

Chunki $5a$ hamesha $5$ ka multiple hota hai, isliye $R$ reflexive hai.

2. Symmetric:

Maaniye $aRb$ sahi hai, yani $2a+3b=5k$ (jahan $k$ ek integer hai).

Humein check karna hai ki kya $2b+3a$ bhi $5$ ka multiple hai.

Chunki result $5$ ka multiple hai, isliye $R$ symmetric bhi hai.

3. Transitive:

Maaniye $aRb$ aur $bRc$ sahi hain.

$2a+3b=5k_1$ ...(i)

$2b+3c=5k_2$ ...(ii)

Dono equations ko add karne par:

$2a+5b+3c=5(k_1+k_2)$

$2a+3c=5(k_1+k_2-b)$

Chunki $2a+3c$ bhi $5$ ka multiple hai, isliye $R$ transitive hai.

Conclusion

Chunki relation reflexive, symmetric aur transitive teeno hai, isliye yeh ek equivalence relation hai.

Sahi Option: (1)