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Prove that a relation R defined on NxxN ...

Prove that a relation R defined on `NxxN` where `(a, b)R(c, d) <=> ad = bc` is an equivalence relation.

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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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