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Introduction to Abstract Algebra (Math 113)

IntroductiontoAbstractAlgebra(Math113) +and ,CosetsandLagrange (ProofsOmitted).. , , (ProofsOmitted).. ,themostlikelyresponsewillbe: Somethinghorribletodowithx,yandz .Ifyou reluckyenoughtobumpintoamathematicianthe nyoumightgetsomethingalongthelinesof: Algebraistheabstractencapsulationofourin tuitionforcomposition .Bycomposition, , , ,evenasalittlebaby. N:={1,2, }, +and . Z:={.. 2, 1,0,1,2,..}, +and .AdditiononZhasparticularlygoodpropertie s, 2Q:={ab|a,b Z,b =0}, +and .Thistime,multiplicationishasparticularl ygoodproperties, ,therealnumbersandthenC, +and , +and +b=b+aforalla,b Q,ora (b+c)=a b+a cforalla,b,c (setswithextrastructure),ofwhichZandQare definitivemembers.(Z,+) Groups(Z,+, ) Rings(Q,+, ) FieldsInlinearalgebratheanalogousideais( Rn,+,scalarmultiplication) ,alltheprotonsonEarth,everythoughtyou veeverhad,N,Z,Q,R, +and.

Here are some examples of maps of sets: 1. S = T = N, f : N → N a $→ a2 2. S = Z×Z, T = Z, f : Z×Z → Z (a,b) $→ a+b This very simple looking abstract concept hides enormous depth. To illustrate this, observe that calculus is just the study of certain classes of functions (continuous, differentiable or integrable) from R to R ...

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