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Fact: i) For finite sets
Au A2, A3, An.. JA, A, Anda AlA4. _ An
for first coording te ex: Azf 123
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AxB ,53,216)/0,7),2,5)2,5),(2,3)} exi Ax PCA)
When i PM PCADE LUTY 23, 6,{1,2} AxP(4) = 0,2 ,CIQ), (1,41.2),
 (2,43) (2,123), (29) (2012) Notice: A x PCA) JE ALL LIPCA)]
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123 Binary Relations
A&B are 2 sets a EA, DEB, we would like to len cw if b is related to a in some Ifashion
Def: A binary relation from A to B is a subset RCA*B09,b ), 9EA, LE133
•If a bit R, then we write aRb or arb,
which means a is related to be  A binary relation on A just means a binary
We also discuss several other topics like Can you be a social worker with a masters in psychology?
relation from A to A. We also discuss several other topics like What is the meaning of arrow formalism?
We also discuss several other topics like How does the body regulate blood osmolality?
xnx  Z Xú
ex: A set of all GMU students
Bi set of CS courses offered
arb if student a is enrolled in b
exi let af 0, 1,2,3,4} and ~ 6='
What is R ÇA?? want anyo x=y so Ra{ xx): XCA
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lex: same A, and let na "
what is R? grbach .R {xy: Xi Y CAXcy
$ 20,1),(0,2), (0,3),694) Don't forget about the age old question of What bonds need to be broken when water is made?
261,2), (1,3), (14)  If you want to learn more check out What is the difference between counseling psychology and mental health counseling?
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I properties of Relations
Let. A be, a set & R a binary relation on A: Let in denote the relation
un is reflexive if a za for all die A Icij z is symmetric if arb bra day is anti symmetric if (arb.and bra) a = b
ex :(€) i r is transitive if arb and brc, then anc
exilet A IN=£1,2,3,...} .
5 :2 I i) This is reflexive since nen for all ne TN li) If nem, does this imply men? no
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keep A = N, but let a b nam n2 m2 2100 Treflexive? not true for all integers 2n2 2 symmetric? nam Gn2tm23,00 6) men 2 transitive? If n2+ m2 100€ m2+k2,100, then n2412100 If you want to learn more check out What are the similarities and differences between homogeneous and heterogeneous mixtures?
Indt transitive since n2 m2=10) X1 " m?ke?, bulg2462=2
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Defi A binary relation a Creally R) on A is I called an equivalence relation if it is reflexive, symmetric, and transitive ea'l relations share important I properties with a Tex: Let A = 2 and any gob is divisible by 3
creflexive:6980 is divisible by 3, so ang 1. Symmetric? if anp a
bis divisible by 3, but I then so is basallab)
transitive? If aubfb2c, then arc. 9clab4cbc
both divisible by z
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It turns out, ea'l relations on a set are easy to classify
Def: A collection {Pi} of subsets of a set A, is a partition of Aif
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12.A=P, U P2 ... Pino
Tex: for A2 € 0,1,2,3,4}
P= {0,1,2 }, P242, 3,43 Is this a partition?  A = pi Up2, but 2e Pin P2.
1
TP140, 7 P2 = £3,42 ? no since PUP2 {0, 1,3,4%
__2€ If Pl=40,1,23 P2243,47 is a partition
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A
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& X.

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Proof: If 4 =0, then IPAELLO})=320
Suppose now that A&O, A= {21,92 ans 1A)'tan want to cont all possible subsets, XCA
For each it{1, 2, 3, either a EX  So for all elements of A we have by a za
possibilities of forming subsets x SA.00 I Def Let A, A2, ... Ao be rets
i) Cartesian product
   A, XA, XA2 x. xAn is the set of ordered notuples
(a,,a2  ah), where each ai tai In particular if A42.AnzĄ, theo A1:A2:
nAnAn A {(x, y X) i xifa for i=1, 03.
ex: À= R=
then on
real time
IX : (x,x)
Rx2= R2
Re is the cortesian plane ........
ex: RX1R XI2 =1R3
This is Eucliden 3space
here : Bloot!
. . _ ici: 80 both are infinite se
but Q PINCER)
Facts: If A&B are sets...
If Ac B and B is finite then so is A, and TALEB ii) If ACB, and A is infinite li) If A&B are both finite, then so is AUB
in) If Ae Bane infinite, then TAMB Icon
be either finite or infinite
ex: A =
Q
Twides
 B=TRIQ.
ABzdins! er A.BEN
AB=W Def for a given set A, dengte by IP (A)
the set of all subsets of A called the leower sel of A ex: 4:0,!}
TPCA{ es, {1},0; {013} ex B:E123
pie puis,21,235,11,23,2.,33,1235,
. 4.2333 Note PC 04