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Fact: i) For finite sets
Au A2, A3, An.. JA, A-, Anda Al-A4. _- 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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12-3 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*B-09,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:6-980 is divisible by 3, so ang 1. Symmetric? if anp a-
bis divisible by 3, but I then so is b-asal-la-b)
transitive? If aubfb2c, then arc. 9-c-la-b4cb-c
both divisible by z
d
La
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
bad
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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 A-42-.-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 3-space
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