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What is a binary relation in discrete mathematics?

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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),

How many binary relations are there?

- (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

What is meant by binary relation?

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

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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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& 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 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,

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