LATTICE in Discrete Structure

LATTICE: UNIT 3 complete notes as per AKTU, LUCKNOW
Discrete Structure & Theory of Logic

1. Lies Late

dgiruren IA pose, io which evey pout £0.63 q
two element of L hos a est upper
bound (LUBY ana a queda lowe boanch

DB tm à Lakes

Zum cunokol by avb and tas Tolo
Gem ounokel by a4b and klos Meet

poseb— Fautially Cuclurol set, wards op the
pueineiple of patettat onduuing lali,
Potet— A Relatron Rs oil do be partral
tel elation when th can sy the
io Hala peau
Pis, GEDIEER
REF Clad), €202),0% BF

2: Anti-sgmmehie - yy R tontairs 01,2)
ab, Cab) alba) then (211) à not allowed.

5: Transihre = iL R tonfains (1,2) 40213)
then Ik should Contern C3) to modes il
Transihre

Ty 0 get A! follows 0 Poutat Ouduirg
Relahor IR Wen tt Klos Poser. THE
olenotd by TASK

2. Ext For a Ser AZ 80233, Check ij Ihe gellWwwing
velarons Au Pose To
Rie 1011),C212), 038%
Ro = UCI, (212), (3/3), C12)» C4 NF
Rar

L£0 ma Poser, but la Rs coe not

genen lite
A botany crolarol set ct biwally a Lattrce ; bub not
au parttaliy oveteud Ser au dala

El tet Le $1,2,35,303 and R beine relatron
Mg dvsito but, Prove mat 2 u Labio
8 E es CLR) à à PoseT 4
>
every par of elements
op L has an 208 4
3 = a GLB, ira alakre
FR
Hasse dragon

gpaerens Testo ar

{ 3030 % So » 30

Luß o

3. Hosse drape m A passe diaguam 1 a
geoph! Le testes) 7 0 poser.

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Brass ide autor "rare Area

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2> Remove the loop at each enter.
Copa)

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heras 4 Rares

u Arrange each elas So Mar au arrows
point upwards

<. Remove au arrowkeadı.

al Consbuet te Hesse dia guam fr
(8172335 S$)

eg Le), (2<3) ete

Re E (UD, (2/2), (3/3), Cu?) 013) , 021393

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ce

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=> Remove tbrensilime velalten

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e 5

up Arrenge Ll tolge

?
à

© Romove aL arrow Ateos
Nd >
f or 2
1

5. Sa ronsidos SAS $2136} cudlh Metern R, (Gtr)

Copshact me Wer Hasse otiaquam 4
the following -

Se

Skp2- Remove Sey Llop x 7
@

Skp3 - Remove “Transihie Wlahon

IR?) 223 — 123
ab bé ae

®

6. mtg thy oleh EA Cee

ee
Consbuetr Hase Diagtam fer in fallocoing

Az 81918, -..123 ab ip a dives b.

Az 51,23, US 6,4, 8, 9, 16,11,12%

12

ES

lo

A al
=
1

QA Puaw hasse Dragua for the following

(1248 10,12, 20,253, /)

ES

Hasse Magram

7. 6 Fincl Component of Poser fer fellocomng

Hasse Dragram—=
Maxime! — b, td
Minimal - a
Greatest — Mo qwateat eme
[Onique)
u lest - a longue)
d e
Maximal - de
7 Murimal - a,b
Gialat 7 No graluk elenet
testo _ MO: Onrgur elernedt
a b
d
a a>
b e
e
à
Maximal d
u dl Mund = à
anime mee Great - of
Hatat — lost -
A bm = Lost eme 7

8. Mend maf - 3h

Maxima! 5
Minimal — A Minimal —2b
Gyrecdet — No Unique Greakat — No Unique
eumet elemen
leak — a Leosl — No unique
elemat-
QE lides Qleroani Of PORET Fiesta
82 Find Meximol, Minimal , greatat and déast
eumet of Pose T5 214,/5,10,12,20,253,/)
SI For Hasse dimgram aaa Ay —selurons
Matimel —12/20,28
Minmel - 25
Greokat - No Unique elem
Least No Unique element
: naw Hasse draquam of the follow
ga > qn y Pe

ay (41213, 4,6/122,/)
» (81213 46,93, 7)
lust, ©)

9. 9

u

@Z check the given welahen à poset er ne %

(nen puotudl to Hasse oliagrem. Az Turn

RECIO, (4232013)

3,3), 3/0, OH

99 POSET Refinar, Anim

4

x
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Bf

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in upwand diagsem.

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213) — (113)
au) — Qu)
34) — A

© termos
errrow
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ihre

Hasse

OW), C22) 0212) A2),

'

14,

Bernat

|;

chagran

10. gt Conde SHA = 81121, 6 won man R,lye)

|
R= fou)

UD ROUE) 02,2) M6), (3/3), 03,6), 16,4) Y
duau haste diaqam.

a
A Dg 2 Bmow Ser -Iop
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a

> © Le

6)

Remove Trunache pP Move un Rernové
42), (22,9 — (1,6) Upuend aro
4 Lemos Ka

013), (310 => re edgy A eu
6 £

3

© 2

7

(in Ww)

[07]
gle praw Hasse diagram of Ou)
Digs 812,3, 6, 91183

WA

6 Hasse oagrem dots not certain any
tes U

©

11. PG Daaw Hasse of Mw followings

ay RE EON, 429,003) CIS,
(212), 0212), Cid), (2,5),
(3D, GA), 8,9), (40,5), C504

Pomove Belt 10PS

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genera RARE
ape 2 :
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k 133

12. b (co, 1)

2 q divin

o assi bibi aelalico

F121 ans 6,10, Pris, 20,30, 603

Veo

QUE check mu following Hasse diagram u Labia

ce not

20 La dad

7 For every pair of eumenk y &
hos LUB 48. Hence, we
gun toy Mal" Lu Lali
pir wise, La not a Lali

Via b © de f
ayo ce ed ef
nl bea et
Plc cc det
ala ad att
eleeett-
glee tt -4
vp per Ma element .

fear and fde
dow aves net est

13. má Hasse

PoséT with
ov nd

Hu

whelhau

cx belenine

ge

teat ce

cliagram u

oh

given

¿

alseacy

11? find ¿on tabu, — Join

i> pue Lois tab. — mer A

iS Hasse aragrem

v

ee 2-2. .
van£oce
e

Seo
a.

Sa +. cos
CUY RA x
ES

6 kus yee ne

Vlabederah

TIL

14. as fi ‘
ind cut ty follewing Shruckwu à fax Zaltie?

a a b
D JA Oy :
B
€
a e à
b
e 4
+ a pri=2
avb=? (not ounel ) =?
2) en

dotsn'k exist

Pas Ta £

a
ev = Net exist
= Net est

art
D E a»
à
e
b ®
H
byes
ane=!

(oros =comeciren)
EZ lod bound — qutala Lou
Que Upper bound — Host nana Belt

Lakfa does mot eur

15. Boundtd Latics in a gpecial

poundid cake 1
pe og allée
on tat fat Said lobe Mount salle", 1 1
Ont Me Dein giralat element 4 a least einmal
in baue lie, à guealat Omen à danoled
Pax À Got etument à oltnobel bu ©

taire weedee Va

Exampu\ (PES, urn) Y e bounetel
d'u une least element of PCS)
sabe greatest eument of PUS)

S- par, (3
PO) 10,103,103, SCH, 10,63, ja,g E49, 090)
3
0 al

Othe

a
IA
8983 ed
w dez
¢
— host
Lumen

(PE), 0,0) à a Boundid talita

¿) u not a Bounckel labre,

Exompu 2 Cu)
Sino, SE hos aim
A — least eurent 1, but
Tastatur element aueh exist
i
4
4

16. » ft
2” |
e le
à
Lakre
D e
= See
a
anb=?
dourt erst
pod €
G
b
am à
ave”,
eve?
Lalo X

LE )
ñ
\ ya / {
N/ o
V
Lohr“ Lolita
19
à
d
b
d
h
4 D
e 5
c
» d
2
(bd) met =
Cha) zen zh
Lab p—

@ |

17. ae upp et bound ef laNece (Maxon) a) sna labtop,

He mas Aus on eme! ‘a, Such inal Maths

ORI, then Tu Calua upparbound a the tati

2 lou bound x lalre (Minimum) (0) 1- in a tar
such mat Va CL)

YE au ous on etemet 00,
(ORA), men OU talud tour bound of Mr talla

Ip} wi) hi) ry)
e t 4 :
4 2
à ‘ à
PS € o
PA 2 :
, :
a b a I
vr 4
rel lala Gouneal lali 4 -y x
O Mo- 140 eus Ne T-d =! ae
o-a

tompumental Lau + À altre £ ù Said lo he compumentec!
if evoy ¿amar Yo cL musk have ar Kosh on complemen!

Deshi bultile lalira — A latice it Solel to be dishibuive if
o) avtb4t)=(avb)A Cave)

MM ob,c EL.
D anlove) lento viont)
Y (abmest Compleat ) <>
anb=0
<p»

avb=1

E Pa ®

18. ee see

Paopukes ÓN | Coin v)

D he Telmpoknt law avaz=a
ana za

2) AssouuHte law —
Associate RE = @vbyve= avthve)
Abaca anthre)
3) CormmutaHit law

avb= bva
oAb= bad
d
b ep bubs 108 Chiat) ab
bab= GB (ab) => b
a CTolampoknt law)
“
3
Assewaue ta
O bvéave) = (evd)ve
E E vin = fave
>. a ES
o @ ade = batane)
o ant = bra
a = 4
commulevie lew Fa

i save ‚e
uy bvd A

19. Bis a OO ae à
Medudau latte A (aie (2,6) U Amd te be A

modulo y it avCbac) =CanbsAc whenewn asc Vabteà

Qu e
Le 4
a
for ony 3 emt yh Bh | pr ony eltmut ance
ov(bno) = Gvbne avccae) =@vone
ova = bac ave = cae
A = a e e

Compu Lake - A Laie à Cal Comput ij cach
Op ik mon- empk Subseñ her a Lost upper bound +

Pa qual owe bouncl

Gaz WE L= {213,63 „Amen Ma lala (41) À Comper
€

1720727
| tres 2
Sa= 813,63

sab-ai@ - Consiam a non-emply subsel Zi op a Latte L,
Then uw called a subte iia of Ly 4) ils À à Lalit,

He. ma opwotico of Li eve ly and a4b6 A who ae,

ca sol) 9E12:3,5,5, 10,15, 309 = 055

j 1. 81/2, 6) 0G 5: 211236303 mn
2°45 ,15 305 6: E13,c, 303
B $US 10, Boy E 1° fr
Y 2,6, 107303 à y
AS orp hon — i

20. Boolean

gebiet

Boolean algebra wat invenled by George Book in 1854.
Boolean algebra D wid to analyze 4 sept Simpli[y te
cbr gilt Cog) circule I WU only ine binary numbat

Ne Lente sta abo call a Binary algebra

AND + oR + not AA
aA} B&B A E f
0
ele ie 10
i 0
1 1

o
1

Laws in Boolean Algebra y

p Coninutedee Tae Ar

BA 5 A+B= 840

2: Assouahve la - (AB).c= AB.
(0+0)+c= A+B tC)
Dikbibate law - Arate)= ABTA

A+B O= (A+B CAFO

us Absmphon law - n+(A:6)= A
PCA +8) =a

s- demargens law - cam)! = Axe!
PERRIER 1
(era! = Ale
6: Twin law y (goypubealatwe) (AN! A
Tomaten Lau Inyoluuhen! at

Fo Tdempokntiay - Ara=A, ARA

$ Complement Jaw — Am!

AA

21. BNP Gals IR Gates Nol Gala

DS Don ni

NS

Sum ep Puodudi1 Tha jormed by adeting COR operation)
A Thue Produch Heim or alto
Call oS ‘make Mun toma eu wepreinkd Lin Um,
they aur me produc (AWD opmaken) of baleen Variables
Shen in normal form oy Compumankd form
Dfin Sor= 2m(0,3)

fu somq muntomo 4 munlnms

Aso, A=0,c=0 Minka Ai Bc!
ei Minka ABLE

asl, 6=0,
sop= ALA, + ABS
D x(s0p)= Eml! 316) cil
= = Pe ow Al -0
Pool ster sito o liga
nee me'c Anc
X(SoP)= Al ale + ABC ABC!
x

Puodud of Suma r FE à fermud by mn plying (And
operators > m Sum toma. Tht Sum toma ou abo
call as fauytounma! » Mar-lams are epotrrdo win
IM! may oma Mu Sum COR opeushon) of Booka Vamaëls
esten in normal fern or Compu em

BE Pos rim (12) (Han Pa product of moxtam 142)

B Be

oo, = Zow-A-0
Hype
EF. CE ren

god Fin Pos= (A +RHd)-CAralie)
O

+: pe.

22. PX (POS) = Mn (0214, 8/7)

= (01090 + (ABAD » (041090) ha)

. case)
BBC FI Minkom Pe
© 00 © mad mo <a
oo + © aisle m Fiz a'acl à age and 88e
NE ma+ms+m64m7)
01 0 1 MR pis ez mansmony
o À | 0 ee MS Fle 302181617)
1 0 0 0 AR mg
vo 1 À pee ms
pro dt age mé
yo. As m]
PGS Fam ı-
F2 ea gra (HAG +21) CE +2) (4442)

p2= (Mi M2. mu My)
p22 HOMME Mu, Ms)
F2: 0012345)

yoz fa Maxkams
o 0 0. + WA tz
0 ° | 2 ML n+g+zl
o | 0 1 “a
o 04d 2 my alar
yo 8 2 mS tz
1 © Of 4 Me
N 0 1 mi
Pe i ms

23. ooolton alqebr teehquts, Smpup ma

AB+ALBAO + BR)
CBtstet bus ut law |
>)

(8.0=8)
(ARB+A-8=48)

& Using
expression’
SL ARA Hera +8.648-¢

= ABFAAB+AC4R ABC

= AB+ACIB+B-C

= AB+A-c + Bley (B+RcC=R)
= B+AC CAB+8=8)
Karnaugh Maps ı—

e Oy © xt .

INTRANET
x lo \ oil és) e ao| 0 [tale
wife [s[+]8 aller $
ES uf [es beler
2-3 ME
AS
o
& N lacy + A'Bcd' + Ach
E plein! a TA + AlactS +ABcd ABC)
ages +06 !cb ae ol tl to
08
nico 16
Fe Aa ab ABD u
10 CB ABD

24. fe (P/O, 8S) =Em( 0,25 Ti 9,1) ¥4(3,8, 102,44)
eS, 1
pa \og ol 1] RP lol
so = 3 ot
ol ly p'as
Uy [ie |:
= gl Pas+Pal
vr Pa! oes
Supe j
Neo Nee a AB AR,
5 A RE a o
Vale E D) th,
E,
> 2 2lo,1/23,4114) A
Fe Alege!

£4 1.PCAB Cd) =3(0,42158 10,11, 14,15)
2 Fonda ayala aula + ny2! +ay2,
3- fCobitd)= a'ble! + Wed! + a'be'd + ab!cld! +
abld+ eed acta! tabed. ®