InterviewSolution
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InterviewSolution
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Prove that a relation R defined on `NxxN` where `(a, b)R(c, d) ad = bc` is an equivalence relation. |
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Answer» R defined on `N xx N` such that (a, b) R (c, d) `iff` ad = bc Reflexivity Let (a, b) `in N xx N` `implies a, b in N implies ab = ba` implies (a, b) R (a, b) `therefore` R is reflexive on, `N xx N`. Symmetry Let (a, b), (c, d) `in N xx N`, then (a, b) R (c, d) implies ad = bc implies cb = da implies (c, d) R (a, b) `therefore` R is symmetric on `N xx N` Transitivity Let `(a, b), (c, d), (e, f), in N xx N`. Then, (a, b) R (c, d) implies ad = bc ... (i) (c, d) R (e, f) implies cf = de ... (ii) From Eqs. (i) and (ii), (ad) (cf) = (bc) (de) implies af = be implies (a, b) R (e, f) `therefore` R is transitive relation on `N xx N`. `therefore R` is equivalence relations on `N xx N`. |
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