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sols8

# sols8 - 20 8 September 22nd Equivalence classes 8.1...

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20 8. September 22nd: Equivalence classes. 8.1 . Describe explicitly the equivalence classes for the relation defined in Exer- cise 7.2. How many distinct equivalence classes are there? (This gives a description of the set Z / in this case.) Answer: The equivalence relation defined on Z in Exercise 7.2 is a b ⇐⇒ a and b have the same remainder after division by n . There are n possible remainders: 0, 1, . . . , n - 1. The key observation here is that if r is one of these n numbers, then r equals the remainder of the division of itself by n . Thus, no two of these numbers are equivalent. On the other hand, if a has remainder r after division by n , then a r , by the same token. Thus, every integer is equivalent to one of the n numbers 0,. . . , n - 1. The conclusion is that there are exactly n distinct equivalence classes modulo this equivalence relation. The quotient set Z / is { [0] , . . . , [ n - 1] } . Here, [ r ] consists of all integers which have remainder r after division by n .

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sols8 - 20 8 September 22nd Equivalence classes 8.1...

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