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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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