Proofby Contradiction Notes is theorem negated RIC We proceed bycontradiction State an assumption Assumption Instatiate a canterexample given negated theorem Any Any appeal Correctderivation of a contradiction Concludes thatcontradiction means original Written in English notjust symbols to definition theorem etc algebra is correct theorem is free is correct and appropriately applied RMAted by contradictionin thatmeans Assume negatedcaption Then ita counterexample By correctappeal But Therefore since contradiction reaffirm original in the case provided theorem to definitions theorems letc correctderivation ofa contradiction using theorems conclude the this assumption led to a contradiction Use specific case when theorem involves for all X Number Theory Notes Factors a factor ofbiff b can be evenly divided bya That is for some non zero integer ak b factorization of a that thereexists a numbers greater than 1 is divisible onlyby 1 and itself thatall factors are prime andthefactivization is wes Aprime is a number greater than 1 orization Fundamental Theorem ofAnthmetic ratoga For all natural such Multiplicity of factor number ofoccurrences SeeCommon Divisor Acommon divisor ofa and b is a number thatdivides them both The greatestcommon divisor ofa and b isthebiggest number thatdivides them both elyPrimel Coprime Two positive ifgad X 4 1 Numbers Represented by G which integers greater than or equal I are relatively prime unique stands to for quotient represented by All xGQ iff x 6 where a G2 and bE 2t numbers rational can be a quotient oftwo integers bycontradiction 1 We proceed 2 Assume the opposite ofthestatement 3 By 4 If we the definition take theareample Elaborateifnecessary 5 However provide contradiction then algebra Example Formatpoints 1 We proceed by contradiction that 1SwEx31 1s1 1 xts knew thatfor any sets 3 By Aand B AVB1 AHHB 1AMB1 This means that the definitainofunion we the contradiction Our cardinality ofSUENY is 15 1 0 1517 assumption 4 Therein lies the states thatISU2x3 is Is butwe have justderived thatthecardinalityofSUSY is St S wecan 51 x s that ISUEX3I conclude
View Full Document
Unlocking...