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Lecture Notes for MATH 455 - WI/Modern Applied Algebra at University of Hawaii at Hilo (UHH)

Notes Information

Material Type:Review Sheet
Professor:Staff
Class:MATH 455 - WI/Modern Applied Algebra
Subject:
University:University of Hawaii at Hilo
Term:Spring 2009
Keywords:
  • Provided That
  • Either...or
  • Ordered N-Tuples
  • Proposition
  • Propositional
  • Isomorphism
  • Cartesian Product
  • Natural Number
  • Natural Numbers
  • Composition
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Sample Document Text

Math 455 Spring 2009 David Ross, Department of Mathematics 1 1 Review of set theory 1.1 Notation Some set theory notation notation: 2;fx : g; [ ; \ ; ; ; ;(;n; ;;;4; {;P ()::: ATB; ASB TA; SA (a;b); ha;bi A B :=fha;bija2A;b2Bg 9 is shorthand for \there exists", for example, 9x such that x2 = 2 8 is shorthand for \for all", for example, 8x;x2 0 Proposition 1.1 hx;yi=hz;wi if and only if x = z and y = w De nition 1.1 A (binary) relation is a set of ordered pairs. Notation: Let R be a relation. We often write xRy instead of hx;yi2R The domain of R is dom(R) :=fxj9yxRygand the range of R is ran(R) := fyj9xxRyg. Note that R dom(R) ran(R), but that in general the inclusion will be proper. R 1 :=fhb;ai : ha;bi2Rg. (Inverse of R.) If A a set, then R A:=fha;bi : aRb and a2Ag (Restriction of R to A.) (Note: sometimes R A:=fha;bi : aRb and a;b2 Ag= RT(A A)) R[A] := ran(R A) (Image of A under R.) If S is another relation, then R S :=fha;bi : 9c(aSc & cRb)g (Composition of ...

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