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Let R be a relation on the set N of natu...

Let `R` be a relation on the set `N` of natural numbers defined by `n\ R\ m` iff `n` divides `mdot` Then, `R` is (a) Reflexive and symmetric (b) Transitive and symmetric (c) Equivalence (d) Reflexive, transitive but not symmetric

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(i) x divides x i.e., x R x, `AA x in N`
`therefore` R is reflexive.
(ii) 1 divides 2 i.e., 1 R 2 but `2cancelR1` as 2 does not divide 1.
(iii) x divides y and y divides x implies x = y
i.e., x R y and y R x implies x = y
`therefore R` is anti-symmetric relation.
(iv) x R y and y R z implies x divides y and y divides z.
implies kx = y and k' y = 2, where k, k' are positive integers.
implies kk' x = z implies x divides z implies x R z
`therefore` R is transitive.
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