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MASON / Math / MATH 125 / What is the meaning of binary relation in the context of discrete math

What is the meaning of binary relation in the context of discrete math

What is the meaning of binary relation in the context of discrete math


School: George Mason University
Department: Math
Course: Discrete Mathematics I
Term: Fall 2019
Tags: Discrete math and Discrete Mathematics
Cost: 25
Name: Math 125 Week 2 Notes
Description: These notes cover what was discussed during week 2 of classes.
Uploaded: 09/06/2019
6 Pages 101 Views 2 Unlocks

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What is a binary relation in discrete mathematics?

R?, {0}}


hafif ksu, x3,0) 49 262412

Fact: i) For finite sets

Au A2, A3, An.. JA, A-, Anda Al-A4. _- An

for first coording te ex: Azf 123



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)]



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?

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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


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?

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C23),(2,4), (3,4) 7123


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

lúv) asb, boc tasca


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


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



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


;مn ي م ۱۰


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.


TP140, 7 P2 = £3,42 ? no since PUP2 {0, 1,3,4%

__2€ If Pl=40,1,23 P2243,47 is a partition












& X.



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


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 =




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

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