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definitionofZ - Math 230 E The Denition of Z Fall 2011 Drew...

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Math 230 E Fall 2011 The Definition of Z Drew Armstrong Here’s a joke definition of the integers: Z := { . . . , - 2 , - 1 , 0 , 1 , 2 , . . . } . We all “know” the basic properties of this set because we’ve been fooling around with it since childhood. But if we want to prove anything about Z (which we do) then we need a formal definition. First I’ll give a friendly definition. This just states everything we already “know” in formal language. As you see, it’s a bit long. Afterwards I’ll give a more efficient, but more subtle, definition of Z . Friendly Definition. Let Z be a set equipped with an equivalence relation “=” defined by a Z , a = a (reflexive) a, b Z , a = b b = a (symmetric) a, b, c Z , ( a = b AND b = c ) a = c (transitive), a total ordering ” defined by a, b Z , ( a b AND b a ) a = b (antisymmetric) a, b, c Z , ( a b AND b c ) a c (transitive) a, b Z , a b OR b a (total) and two binary operations a, b Z , a + b Z
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