By Carol Whitehead (auth.)

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D) The set of animals in London Zoo; x ~ y means x andy belong to the same species. (e) The set of paintings in the National Gallery; p ~ q means that p and q were painted by the same artist. (f) The set of students enrolled on any degree course at a given university: s ~ t means s and t are enrolled for the same degree course. 1). (b) ~ is the relation defined on Z by x ~ y means that 1xl = IYI· Check that ~ is an equivalence relation on Z and find the equivalence classes. 1b, 1(e)). Find (a) (1, 2); (b) (0, -3)); (c) (a, b); describe this set geometrically on a cartesian diagram.

4 CONGRUENCE RELATIONS ON THE INTEGERS In this section we introduce a particular type of equivalence relation on the set Z of integers. We shall first give a formal definition of divisibility of integers. Divisors and multiples Let a, d E Z. We say that a is divisible by d (or that a is a multiple of d) if there is an integer k E Z such that a = kd. Then d is called a divisor of a. 1 1. Let 5 be the set of integers divisible by 4. Then 5 = {4r: r E Z}. 45 GUIDE TO ABSTRACT ALGEBRA 2. Let P be the set of multiples of an integer m E Z.

They are either true or false. Sentences such as 'n is an integer', 'x > 3' etc. are called conditions on the variable n (or x). A condition may be true for some values of the variable and false for others. For example, 'x > 3' is true when x = 10 but false when x = 1. ' The following are alternative wordings of the same statement: (a) (b) (c) (d) If condition p holds, then condition q holds. Condition q is implied by condition p. Condition q holds if condition p holds. p is a sufficient condition for q.