DefinitionofZ - Math 230 E Fall 2011 The Definition of Z Drew Armstrong Here’s a joke definition of the integers Z:= 2 1 1 2 We all “know”

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Unformatted text preview: Math 230 E Fall 2011 The Definition of Z Drew Armstrong Here’s a joke definition of the integers: Z := { ...,- 2 ,- 1 , , 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...
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This note was uploaded on 01/08/2012 for the course MATH 461 taught by Professor Armstrong during the Fall '11 term at University of Miami.

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DefinitionofZ - Math 230 E Fall 2011 The Definition of Z Drew Armstrong Here’s a joke definition of the integers Z:= 2 1 1 2 We all “know”

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