25 - Examples 1. Given n ∈ N , the binary relation R on Z...

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Unformatted text preview: Examples 1. Given n ∈ N , the binary relation R on Z defined by a R b if and only if n divides into (b − a) is an equivalence relation. 2. The binary relation S on the set Z × N defined by (z 1 , n1 ) S (z2 , n2 ) if and only if z1 × n2 = z2 × n1 is an equivalence. 3. Let A be any set. Then the identity relation id A : A × A is an equivalence relation. 4. Given a set Student and a map age : Student → N , the binary relation sameage on Student defined by s1 sameage s2 if and only if age(s1 ) = age(s2 ) is an equivalence. 5. In definition 4.22, we define the relation ∼ between sets, which characterises when (finite and infinite) sets have the same number of elements. Proposition 4.23 shows that ∼ is an equivalence relation. 6. The logical equivalence between formulae, given by A ≡ B if and only if A ↔ B , is an equivalence. The above examples suggest that equivalence relations lead to natural partitions of the elements into disjoint subsets. The elements in these subsets are related and equivalent to each other. D E FI N I T I O N 3 . 1 4 ( E Q U I VA L E N C E C L A S S E S ) Let R be an equivalence relation on A. For any a ∈ A, the equivalence class of a with respect to R, denoted [a]R , is defined as [a]R = {x ∈ A : a R x}. We often write [a] instead of [a]R when the relation R is apparent. The set of equivalence classes is sometimes called the quotient set A/R. In examples 1 and 2 just given, Z /R represents the integers modulo n, and (Z × N )/S is the usual representation of the rational numbers. 26 ...
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