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=
itt (Set, Relation 2 function)
__A___.wett__ define __ collection of clase or
a bE. is called a aaa
Include ip the Bob
es _
bn SP ol Ae Ree eee DEE Ee.
-2 -lal= Whole. a ae
Zot = Th ESAS
= Ration A NPE — ste sia
R= Real “Mo.
on Comple No Cx tiy)
_k_ Represent _-
i. Tabulow __form_ox Roaster fom yo
CE. ET ore ee a
2. Set _puilder [Property form
A= tx xs 4 £ 02
¥ Dillesont bypes.-of Sets.
1 Null Fempby 1 I Noid Sot. ati
3 Finite Sop nies seh
Tae ieeb
A Tnfinite. S goth a ee
(iron aS ora alte ee
Ps, Subset = lok A pnd Bion oh sit
- ~ every __elernent of Ais also——
_on___elemont nf BR. then" A_is___ called
_ Subsek of 8 Mia a
nce [aca yxcal i ne Fl - Set — =
7
o Proper “subsok__ = ee ie
_ond_ ne 16 sn 2
————_.__Seb__jhen_A__is called
Proper. “Subset: ol ig ihe
exist atleast ‘one _eloment 8 5
—_——__lohich.._does- noat_¢€ A at oA AGA SL Ha fiat, fish,
tat ee es
_# Operation secre
—L_Union of ge a i SB
bed Da oa ee ee Y)
A= thea eh aies at a
——Drample = oy hie s fo
Sa ee Tee i a
se ee la eat Ss oe
ean
cede Eat on phe
2 5.__Digeit. a melee nO mey ad
oo KOE ES Ear EGE Ae -Diference “al. of hoo so
eal Sel anol tah ad
prego Me This Set.
+ Paoperties Nae Set perce
pa Commutative low = = >. Associative — tow (4) 8) equal
Page = gy | Av( ue) = = (Aua)uC Leo d.
(b) A (@nc)= an) ne
3__Tiempotent to ae eae
reer hice eae
BSE Annee we
_4__Dishibutive S ‘lh
Anfang = (ans) Canc) :
a Av (ane).=_ (av) aie
+ Difference eee De Heagon i
A= (ane) = -8)uA- ¢)
~ n= LB0e) = (AB) (0-¢)
eRe IEE (Aue) = Ane
Be) ane ee
= x_€ (Ave) : =
Re
0
St
: ae ees Pes nde et
Xe Ane!
tA GY xd ea
—-{Aua) fang
ana’ £ ‘Cava
€ poe
X EA ond x 6B Coe
x4 A onor x ¢ 9 og
Tom eq YY and (9)
(Ava) Now
Poot An (auc) (nnsyu (anc)
Yo Pool AN(Boc)_ ¢ Cona)u (aac)
lot
x © An (Bbc)
x © A cid x © (Buc)
TEN ond (NER oF KE 3) ‘
(XE A andre B) ov (ren and TEC)
x © (NOB) ov x € CNne)
x (nna)uCnne)
An(Buc) & Chine) u(anc) ——())
vo Proof (ANB) UCANc)_¢ AN (Buc)
ler
% © (ANA) OCANe)
| = x €(nne) ov xe Canc)
XEN and 268) oF (* €A and x €c}
x (Nand B) om x © (Mand)
xe A and (Roc) :
xe An( Buc)
" 20\9}2002
+ Principate _ oh inclusion __
~
ne of 09
(Ana)u(anc) £ An(avd. — &
—Taom WY and GL nee
—_antave) = fone) on es
om ‘exclucton. aan
ea eset a
~ tang) = Tf
ms | Avsve! =
[AL+ IBl4_| \c=lansl — tact lind
— + | Ansnc)
aon) erg we Ol eee os
: res ae “lel = Ianpl —— tae
a a Be _Bucl = Janel
=-_Jal+ Iel+tcl- lencl- Janel
= _\Al+ I8l+ leL- \encl- lantevol
= ial+ \eista- ~\8ncl- -lianenu taro
pied Ge GA Lane) canes] a
=" _Jal+ |l4 Icl=_Iencl- Hinnets + lane =
= _Jpl+ jelt tcl=tenct~ 14 nats lane!-
anne cee ay
= lal + lel + bet Ipnct = INnal~ Tan!
= * Tangne!
—Nenn— Diagsam-_
—CA-B) ss
> 1A Aa APL = Als Leffel - LArnpsl + La.aana, 1 Ordered Paix (a,b)
9
Cortiscicin — Product of — two Sets
Ax (8c) = (axa) n(Axc)
(ee eA (Cane
TEA, NE C8nc)
XEA,(YEB ond. Yec)
(XEA, YOR) and (LEA, VEC)
XY © AXB and (xy) © Axe
. (ay Lax a) nCAxc)
Ax(Bnc) © CAxa) (Axe)
lets Gay) © Bara) ptaxc) aotet ‘
xe (Ar mY ecax) vo
and (YEA » YC o)
(x,¥) CON¢ )
Va
ee (au) € fy
| Cod © vx (89
(Axa) nfaxc) © Wx(ane) — &)
From. &y- GW) and (2)
Gs Se NaS 24\og|22
13
_ Relation ~ “tek A and B__be- hoo
ee ie a a
is Rasch of @ cross “AAXB denoted -
hye pe
= Ate x
eee ee ae
_Ex= A = tia.s,9f
“JaxBl= ie
ge 1 oo4,af isl= 4
i
AB. 100), €12),-0,4),0,8), (a0)
(9,9) (3,4), (4,8), (5,0). (5.2),
4, 5.8) 9,0),,-(92), (214), (eal
ST aa, Ga Gal
pes Bah Domain= 14asho
- Range. = 19,4, eps SS 2 Ay
+ merce -sielation = 1 _is_the_ _telotion_
_ oe - +e get A tp
Bet then the RR" ig_a relation
Bo to A anen “denoted. by a
-Compasiion _-elertion - of hoo relation. :
oR = 3 they exis an — _ eloment
Ses oa —tohich € @ — Such thot —
ot Era ‘ond. Coc-€8 $ a
ce a
a 2 toa Saas te) a : 1
SoR = 1 (ac) = an (1.8), (3,0) |
“Bi mary ~ Relation. =
Ex. Ah = tinal
xn,
[oe "Re et Set ylation” In Go & then
Ree lice ae — Reflexive then every. element
Of. Set A ot Related to self.
pe et | 0) & Re g 26)00)22
eS — its
ye _ Symmetric = a a, 6 eR “Ob! a R_
as —aRb_ > bRa a
— k= Then, an) ota a, a, A at
_ “Transitive _ = Ss
eS (lent and (b,c) ¢ eR => > acd R_
an Pep ee S. = lnbe del ne ey
WEE
a =
=i Re {Goyb) oid, Cand, Ledde Ce) —
at
4. Ant Simin EERE Ra “relation
: po ino Sok A then
ae called _ “anh. Symmelsic if
Es Leb) Rand C0) Ta ash
oe ——#&=—ta Q=b and b+a ae
= _aShb and. bea et
xT dentity Relation —
Th Ris velation— in
Ae ne kis
Called iclentity Relation if
(Qn) GR = ash V abe R
R- A= X23} :
R= JO 2 bod, (3.3), G3), (az)t Ref
RSG ol ee Te 8
R= f(t), (9,2), 03, 3)} Retev_ Tde-v
Equivetlence velettion = “TT Ris gelation
: in__ get _B
then Rs calleel
if
equivalence,
Retleai ve , Symmetric + Wansitive, 2 ]o8]22
Z ee ae — 18
~) “Topic —
aes Demorgan's Jouo
= Explain. — Noviouwr algebraic 4 exh
See _ Operation J Prepesty—on ee
= State and Prove Disttibutive edu a
= ar -No- 6 PNo-95 (145 Stucente)
* Function — ox a Eee Some
“Th function. —, oie 1k. necessary ne all —
S-sselement of. Got Aus “tet S getateds
lo js. Set B Should be 6 apt * Deflesence bho —Aunction — end wteletion
Tn relatidn . if is
—_AlL_element. af ice
Set A Should be _ ——Nat_ecessarry ty
— —mapped_loSet Boal eloment sh
a . Sot A Should be
‘ eS “-anapped oo ea
+e Function. ae
Tame i ee De
Oe he On One-One. into _. Ee rs say eaeTnine
~-—__Many-=one Onte
+ Onin = vonge ob 4 “oill_the a
- Shin — xorg e—of_f Proper _ Subs eb to. Sek B_ =
enue ee fay
oO
‘Vay: toy One
fon)= foe)
Rares Baga
CLS 2. One - One.
Onto- let Nu R 89. tha Ys flo)
N= log :
As Fa)
eee E
PCat) = log ua loge,
i
1
1
‘ety
Ex- Show thal dhe
wheve CR). =
Mapping do pop
ne is one ~ one
ew
onto. : Xo, a _ . 7
aco ae ee ees
eee Xa _ i aa
Ge tae Sm saa One- One =
Sok eres ee ee eng
a ‘Nes _ { a
——— — X coms —
an =e ee ee bt
a a N me eee
flA)= y Ooie beignst grjog|22
: ee as
ke “Countable ond _
Finite and:
aoe
“oncounlable set
“Antirike set
Tl the cet ie infinite —4hen A
Sis cattod —Countoble if “Hs elernenie
——Con_be_ Pur. ly Feerpp_onto __
rnc eh with the Set of natural _
_-mumbe' . :
+ Real_ “no ia uneountable “gel nna
Pact thot, ine Get Tah aw a ag a
lp __ Countable. is iu) oe
2
fawd= 1G») Gus, #Ou #02)
‘ -Pantial _Onler_ “aylotipn =
2 Reltentive.. Rellexdve as
_Anlin Aymmelic 2
= Transitive: So ee esiehs
~ Composite. function.
AG LTR 4. cons paeieiie
teas ates
= LS HS = 2
fig WS eta entradas”
pes ya
ae ae 2g lion)
oe Maihonmilal eekutina Se 33
Unit- 2
Lge obrate- ‘boil ure
yO
tek Gr he non, ernpiy. Set then.
OX = (rb) 0 € Gb dq. ois
on opesntion on OE AX =
where 0. is The binary.
operation
Srrcup _- An alpetrioie Styuchuve (61,0)
tohore G1 is a “Ate N=.
einply. Sek and. 0 binary wis
0 aie defined on ar is talled
ol it the operation..o
salicthe
We ‘oltovoinr postulate
ond — ( Gxiorne) 7
\\G, Closure properly = Tt “Of ‘ond fb _
E G_4hen
noob. € ee
ab€G > Qohe G, Va,b €or