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Please Solve RD Sharma Class 12 Chapter Relation Exercise 1.2 Question 15Subquestio (i) Maths Textbook Solution.

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Given: $Hence\: proved, R\cap S \: and \: R\cup S\: are \: symmetric.$
Hint: For symmetric relation.
$\text{a=b is true then b=a is also true.}$
Given: R and S are relation on a set A.
R and S are two symmetric relations on set A.
$To\: Prove: R\cap S\: is \: symmetric$
Explanation:
$Let\left ( a, b \right ) \: \epsilon \: R\cap S$
$(a, b) \: \epsilon \: R \: and\: (a, b) \: \epsilon \: S$
$(b, a) \: \epsilon \: R\: and\: (b, a) \: \epsilon \: S$
$(b, a) \, \epsilon \, R\cap S \; \; \; \; \because [R\: and\: S\: are\: symmetric]$
$(b, a) \: \epsilon \: R\cap S$
$R\cap S\: is \: symmetric$
$To\: prove: \! R\cup S \: is \: symmetric$
$Let (a, b) \: \epsilon \: R\cup S$
$(a, b) \: \epsilon \: R \: or (a, b) \: \epsilon \: S$
$(b, a) \: \epsilon \: R \: or\: (b, a) \: \epsilon \: S$
$(b, a) \: \epsilon \: R\cup S \; \; \; \; \; \; \; \; \because [R\: and\: S\: are\: symmetric]$
$R\cup S\: is\: symmetric$
$Hence \: proved, R\cap S\: and\: R\cup S\: are\: symmetric.$