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section2and3notes - Lectures 2 and 3: algebraic...

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Unformatted text preview: Lectures 2 and 3: algebraic descriptions of N , Z , and Q . Construction of Z = { , 1 , 2 ,... } . The inverse of the succesor function f ( n ) = n + 1 is g ( n ) = n- 1. Z is the smallest set containing 0 and closed under f and g . Construction of Q = { m n ,m,n Z ,n 6 = 0 } . The inverse of the multiplication map m k ( n ) = kn is d k ( a ) = a/k when k 6 = 0. Q is the smallest set containing 0 and closed under f , g , and m k , d k for all k 6 = 0. Axiomatic approach to N , Z and Q . N has the operations + and satisfying: associativity A 1 : a + ( b + c ) = ( a + b ) + c M 1 : a ( bc ) = ( ab ) c commutativity: A 2 : a + b = b + a M 2 : ab = ba additive identity: A 3 : a + 0 = a M 3 : a 1 = a distributive law: DL : a ( b + c ) = ab + ac Z has all properties plus: additive inverse: A 4: a , (- a ) so that a + (- a ) = 0. Q satifies all these plus: multiplicative inverse: M 4: a , a- 1 so that a a- 1 = 1....
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