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Question
Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.
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Solution
We observe the following properties of R.
Reflexivity :
Let (a, b) be an arbitrary element of Z × Z0. Then,
(a, b) ∈ Z × Z0
⇒ a, b ∈ Z, Z0
⇒ ab = ba
⇒ (a, b) ∈ R for all (a, b) ∈ Z × Z0
So, R is reflexive on Z × Z0.
Symmetry :
Let (a, b), (c, d) ∈ Z×Z0 such that (a, b) R (c, d). Then,(a, b) R (c, d)
⇒ ad = bc
⇒ cb = da
⇒ (c, d) R (a, b)
Thus,
(a, b) R (c, d)⇒ (c, d) R (a, b) for all (a, b), (c, d) ∈ Z× Z0
So, R is symmetric on Z×Z0.
Transitivity :
Let(a,b), (c,d),(e,f) ∈N × N0suchthat a,b) R(c,d) and(c,d) R(e,f).Then,
a, b) R (c, d)⇒ ad =bc (c, d) R (e, f) ⇒ cf =de } ⇒ (ad) (cf)=(bc) (de)
⇒ af=be
⇒ (a, b) R (e, f)
Thus,
(a, b) R (c, d) and (c, d ) R (e, f) ⇒ (a, b) R (e, f)
⇒(a, b) R (e, f) for all values (a, b), (c, d), (e, f) ∈ N × N0
So, R is transitive on N × N0.
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