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Final

MATH 4389 Final: Abstract_Algebra(part_1)
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Department
Mathematics
Course
MATH 4389
Professor
Almus
Semester
Spring

Description
duate thm) Math 4389compact summary of an undergra Abstract Algebra These notes are intended to be ad recall:rmaesugmbaerstsaom number that’s a multiple of 4. Divisibiltthe:ehstsdmgiaital,oidt,is aormutiple of 3.mnu1isaiviiliel,b ful:     .   f and the TBt Ε  , then of    d n uade| SA | and for some   of   and    Sy Sx 1T  f image inverse image if is odd 1 if is even fA B  xx f f       y b n e v i g f fx Determine 0,1,5,9 and For yDour:referis called thecalled theinitions you may find help   1   1 1               f f 2 f f     :  given by :  given by Ex: f fx x Determine 7, 1,0,1,2 Ex: f fx 7,D1,r,1,ne 2, 1,0,1,2 and one as inverse , if and only if , if and only if injective surjective , or onto one-to-one example, every singleton set h fs called et or a set of size 2? fis called . . Then  : fA B    . The function  fA   Is ittutisteit,ir theeprevtoDef: Let x A fucorrespondence, or bijective.one-to-one is called a one to fonto? fonto? fonto? fonto? fone-to-one? I fsne-to-one? Is fone-to-one? I fsne-to-one? Is . Is . Is 2. Is 3 . Is 3 2     xx xx x x f f f f be given by be given by be given by be given by       f f f f Ex: Let Ex: Let Ex: Let Ex: Let fonto? fnto? fone-to-one? Is fne-to-one? Is . Is . Is     xx xxx f f be given by be given by   : : f f Ex: Let Ex: Let . x x if is evens odd 3 x 2  2        fx onto? f be given by  f one-to-one? Is f Ex: Let Is nve sifi xx xx 2 if is odd   be given by gx  : . Problems? Why? g and and x and  if is even : fg gf f ddoxs 2 fi    x  fx Ex: Let Compute , a and of | ca with multiple c if bhere is an inteber denoted by b of and is an integer br that b ab otherwise divisor and | is a a (gcd) of a aivides and 0 f 0 i | . and is defined by | of inteers   cb ab c  , y max : bare integers, then   . We also say that and greatest coabon divisor a b = ad common ivisor ab ab . Th Number T DheforIfReview:enDef: Ais by , or simpl p 1and the only divisors of m not 0. Then there exists a p if dcan be wridis the smallest positive integer that can , and . Also, n b prime integer and is a aand m p  p be integers, at least one of the b b, and aand and  a TheoLemunique gcd Common Divisor):einfire.er . | pb or pa , then either | pa is prime and p Theo Ifem (Euclid’s Lemma) l to the of two integers is always equa GCD Fan:rduMprdfamtplGrLhMmndfves0 and 3234? by writing .  ab : m divides m if od ab m Congruence Modulofine an equivalence relation on the set , .   d o |m  abm  relation (reflexive, symmetric is the set  Nottrahsatthi).is an equivaTenecequivalence class q . .  and if and only if they have the same
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