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Assertion (A): If R is a relation defined on the set of natural numbers N such that R = {(x, y): x, y in N and x + y = 24}, then R is an equivalence relation. Reason (R): A relation is called an equivalence relation if it is reflexive, symmetric, and transitive.

1. A is True, R is True; R is a correct explanation for A.
2. A is True, R is True; R is not a correct explanation for A.
3. A is True, R is False
4. A is False, R is True.

Correct answer: A is False, R is True.

Solution

The relation R = {(x,y): x+y=24} is symmetric (if x+y=24 then y+x=24) but not reflexive (we need (x,x) in R for all x in N, which requires 2x=24, true only for x=12). It is also not transitive: (12,12) is in R, but (6,18) and (18,6) are in R — however (6,6) is not, so transitivity also fails. Therefore R is not an equivalence relation: Assertion A is False. The Reason R is a correct definition and is True. Answer: A is False, R is True.

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