bm cycle 7 session 1
TRANSCRIPT
7/23/2019 BM Cycle 7 Session 1
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PAN African e-Network Project
PGD IT
BASIC MATHEMATICS
Semester - I
Session - 1
Dr. Nitin Pane!
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Detailed Syllabus
1. Introduction to sets (sets of numbers (N, Z, Q etc)),subsets, proper subsets, power sets, universal set, nullset, euality of two sets, !enn dia"rams .
#. Set operations (union, intersection, complement and
relative complement)$. %aws of al"ebra of sets (&'e idempotent laws, t'e
associative laws, t'e commutative laws, t'e identitylaws, t'e complement laws (i.e. ∪ c * +, ∩ c * ,(c)c * , +c * , c * +), De -or"ans laws) proofs oft'e laws usin" labelled "eneral !enn dia"ram, proofs ofresults usin" t'e laws
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Instructional /b0ectives
Illustrate properties of set al"ebra usin"!enn2dia"rams.
3rove various useful results of set al"ebra.
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Session /b0ectives
1. Introduction of sets
2. R epresentation of sets
3. Types of sets
4. Subsets and proper subsets
5. Universal sets
6. Euler-Venn diagram
7. Algebra of sets (i.e. union, intersection, difference etc.)
8. Complement of set
9. Laws of algebra of sets
10.De Morgan’s laws
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Introduction to Sets
4eor"e 5antor (167821918), in 1698, wast'e first to define a set formally.
Definition 2 Set set is a unordered collection of :ero of more
distinct well defined ob0ects.
&'e ob0ects t'at ma;e up a set are calledelements or members of t'e set.
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Introduction to Set &'eory
set is a structure, representin" anunordered collection ("roup, plurality) of:ero or more distinct (different) ob0ects.
Set t'eory deals wit' operations between,relations amon", and statements aboutsets.
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<asic notations for sets
=or sets, we’ll use variables S, T , U , … >e can denote a set S in writin" by listin" all of
its elements in curly braces
? @a, b, cA is t'e set of w'atever $ ob0ects are denotedby a, b, c.
Set builder notation =or any proposition P ( x )over any universe of discourse, @ x BP ( x )A is the
set of all x such that P(x). e."., @ x B x is an inte"er w'ere x C and x E8 A
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+Famples for Sets
GStandardH Sets Natural numbers N * @, 1, #, $, A
Inte"ers Z * @, 2#, 21, , 1, #, A
3ositive Inte"ers Z+ * @1, #, $, 7, A Jeal Numbers R * @7K.$, 21#, π, A
Jational Numbers Q * @1.8, #.L, 2$.6, 18, A
(correct definition will follow)
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Specifyin" Sets
&'ere are two ways to specify a set1. If possible, list all t'e members of t'e set.
+.". * @a, e, i, o, uA
#. State t'ose properties w'ic' c'aracteri:ed t'emembers in t'e set.
+.". < * @F F is an even inte"er, F C A>e read t'is as G< is t'e set of F suc' t'at F is an even
inte"er and F is "rater t'an :eroH. Note t'at we canMt listall t'e members in t'e set <.5 * @ll t'e students w'o sat for <I& I&111 paper in #$AD * @&all students w'o are doin" <I&A ? is not a set
because G&allH is not well defined. <ut+ * @Students w'o are taller t'an L =eet and w'o are doin"
<I&A ? is a set.
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<asic properties of sets
Sets are in'erently unordered ? No matter w'at ob0ects a, b, and c denote,
@a, b, cA * @a, c, bA * @b, a, cA *
@b, c, aA * @c, a, bA * @c, b, aA. ll elements are distinct (uneual)
multiple listin"s ma;e no differenceO
? @a, b, cA * @a, a, b, a, b, c, c, c, cA. ? &'is set contains at most $ elementsO
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Some 3roperties of Sets
&'e order in w'ic' t'e elements are presentedin a set is not important. ? * @a, e, i, o, uA and
? < * @e, o, u, a, iA bot' define t'e same set. &'e members of a set can be anyt'in".
In a set t'e same member does not appear
more t'an once. ? = * @a, e, i, o, a, uA is incorrect since t'e element PaMrepeats.
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Some 5ommon Sets
>e denote followin" sets by t'e followin"symbols ? N * &'e stet of positive inte"ers * @1, #, $, A
? Z * &'e set of inte"ers * @,2#, 21, , 1, #, A
? J * &'e set of real numbers
? Q * &'e set of rational numbers
? 5 * &'e set of compleF numbers
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Some Notation
5onsider t'e set * @a, e, i, o, uA t'en
>e write GPaM is a member of PMH as ?
a ∈ >e write GPbM is not a member of PMH as ? b ∉
? Note b ∉ ≡ ¬ (b ∈ )
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Definition of Set +uality
&wo sets are declared to be eual if and only if t'ey contain eFactly t'e same elements.
In particular, it does not matter how the set is
defined or denoted. =or eFample &'e set @1, #, $, 7A *
@ x B x is an inte"er w'ere x C and x E8 A *@ x B x is a positive inte"er w'ose suare
is C and E#8A
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Infinite Sets
5onceptually, sets may be infinite (i.e., notfinite, wit'out end, unendin").
Symbols for some special infinite setsN * @, 1, #, …A &'e natural numbers.Z * @…, 2#, 21, , 1, #, …A &'e inte"ers.R * &'e “r eal” numbers, suc' as
$K7.16#67K19#979616191K#6197$1#8… Infinite sets come in different si:esO
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&'e +mpty Set
∀∅ (“null”, “t'e empty set”) is t'e uniueset t'at contains no elements w'atsoever.
∀∅ * @A * @ x|FalseA
No matter t'e domain of discourse,we 'ave t'e aFiom
¬∃ x x ∈∅.
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niversal Set and +mpty Set
&'e members of all t'e investi"ated setsin a particular problem usually belon"s tosome fiFed lar"e set. &'at set is called t'e
universal set and is usually denoted by PM. &'e set t'at 'as no elements is called t'e
empty set and is denoted by Φ or @A. ? +.". @F B F# * 7 and F is an odd inte"erA * Φ
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!enn Dia"rams
pictorial way of representin" sets.
&'e universal set is represented by t'einterior of a rectan"le and t'e ot'er setsare represented by dis;s lyin" wit'in t'erectan"le. ? +.". * @a, e, i, o, uA
ae
iou
A
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<asic Set Jelations -ember of x ∈S (“ x is in S”) is t'e proposition t'at ob0ect x
is an ∈lement or member of set S. ? e.g. $∈N, “a”∈@ x B x is a letter of t'e alp'abetA
5an define set euality in terms of ∈ relation∀S,T S*T ↔ (∀ x x ∈S ↔ x ∈T )“&wo sets are eual iff t'ey 'ave all t'e samemembers.”
x ∉S ≡ ¬( x ∈S) “ x is not in S”
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+uality of two Sets
set PM is eual to a set P<M if and only if bot'sets 'ave t'e same elements. If sets PM and P<Mare eual we write * <. If sets PM and P<M arenot eual we write ≠ <.
In ot'er words we can say * < ⇔ (∀F, F∈ ⇔ F∈<) ? +.".
* @1, #, $, 7, 8A, < * @$, 7, 1, $, 8A, 5 * @1, $, 8, 7AD * @F F ∈ N ∧ E F E LA, + * @1, 1R8, , ##, 8A t'en * < *D * + and ≠ 5. 9
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5ardinality of a Set
&'e number of elements in a set is calledt'e cardinality of a set. %et PM be any sett'en its cardinality is denoted by BB
+.". * @a, e, i, o, uA t'en BB * 8.
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Subsets
Set PM is called a subset of set P<M if andonly if every element of set PM is also anelement of set P<M. >e also say t'at PM is
contained in P<M or t'at P<M contains PM. It isdenoted by ⊆ < or < ⊇ .
In ot'er words we can say
( ⊆ <) ⇔ (∀F, F ∈ ⇒ F ∈ <)
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Subset ctd
If PM is not a subset of P<M t'en it is denotedby ⊆ < or < ⊇ ? +.". * @1, #, $, 7, 8A and < * @1, $A and 5
* @#, 7, LA t'en < ⊆ and 5 ⊆
1 35
24
6
BA
C
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Subsets < G is a subset of <H
< if and only if every element of is alsoan element of <.>e can completely formali:e t'is < ⇔ ∀F (F∈ → F∈<)
+Famples
A = {3, 9}, B = {5, 9, 1, 3}, AA = {3, 9}, B = {5, 9, 1, 3}, A B ?B ? true
A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, AA = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A B ?B ?
false
true
A = {1, 2, 3}, B = {2, 3, 4}, AA = {1, 2, 3}, B = {2, 3, 4}, A B ?B ?
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Subsets seful rules
* < ⇔ ( <) (< ) ( <) (< 5) ⇒ 5 (see !enn Dia"ram)
UU
AABB
CC
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Some 3roperties Je"ardin"
Subsets =or any set PM, Φ ⊆ ⊆
=or any set PM, ⊆
⊆ < ∧ < ⊆ 5 ⇒ ⊆ 5
* < ⇔ ⊆ < ∧ < ⊆
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3roper Subsets
Notice t'at w'en we say ⊆ < t'en it iseven possible to be * <.
>e say t'at set PM is a proper subset ofset P<M if and only if ⊆ < and ≠ <. >edenote it by ⊂ < or < ⊃ .
In ot'er words we can say( ⊂ <) ⇔ (∀F, F∈ ⇒ F∈< ∧ ≠<)
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3roper (Strict) Subsets Supersets
S⊂T (GS is a proper subset of T H) meanst'at S⊆T but . Similar for S⊃T.
S T
Example:
{1,2} ⊂{1,2,3}
Venn ia!"am e#ui$alen% o& S ⊂T
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!enn Dia"ram for a 3roper Subset
Note t'at if ⊂ < t'en t'e !enn dia"ramdepictin" t'ose sets is as follows
If ⊆ < t'en t'e disc s'owin" P<M may overlapwit' t'e disc s'owin" PM w'ere in t'is case it isactually * <
B A
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3ower Set
&'e set of all subsets of a set PSM is called t'epower set of PSM. It is denoted by 3(S) or #S.
In ot'er words we can say
3(S) * @F F ⊆ SA +.". * @1, #, $A t'en
3() * @Φ, @1A, @#A, @$A, @1, #A, @1, $A, @#, $A, @1, #, $AA
Note t'at B3(S)B * #BSB
. +.". B3()B * #BB * #$ * 6.
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Set /perations 2 5omplement
&'e (absolute) complement of a set PM ist'e set of elements w'ic' belon" to t'euniversal set but w'ic' do not belon" to .
&'is is denoted by c or T or U . In ot'er words we can say
c * @F F∈ ∧ F∉ A
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!enn Dia"ram for t'e 5omplement
A
A'
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&'e nion /perator
=or sets A, , t'eir union A∪ is t'e setcontainin" all elements t'at are eit'er in A,or (“∨”) in (or, of course, in bot').
=ormally, ∀ A, A∪ * @ x B x ∈ A ∨ x ∈A. Note t'at A∪ contains all t'e elements of
A and it contains all t'e elements of ∀ A, ( A∪ ⊇ A) ( A∪ ⊇ )
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Set /perations 2 ∪nion
nion of two sets PM and P<M is t'e set of allelements w'ic' belon" to eit'er PM or P<Mor bot'. &'is is denoted by ∪ <.
In ot'er words we can say
∪ < * @F F∈ ∨ F∈<A
+.". * @$, 8, KA, < * @#, $, 8A ∪ < * @$, 8, K, #, $, 8A * @#, $, 8, KA
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@a,b,cA∪@#,$A * @a,b,c,#,$A
@#,$,8A∪@$,8,KA * @#,$,8,$,8,KA *@#,$,8,KA
nion +Famples
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!enn Dia"ram Jepresentation for
nion
BA
A ∪ B
35(
2
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Set /perations 2 Intersection
Intersection of two sets PM and P<M is t'eset of all elements w'ic' belon" to bot' PMand P<M. &'is is denoted by ∩ <.
In ot'er words we can say
∩ < * @F F∈ ∧ F∈<A
+.". * @$, 8, KA, < * @#, $, 8A ∩ < * @$, 8A
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&'e Intersection /perator
=or sets A, , t'eir intersection A∩ is t'eset containin" all elements t'at aresimultaneously in A and (“∧”) in .
=ormally, ∀ A, A∩≡@ x B x ∈ A x ∈A. Note t'at A∩ is a subset of A and it is a
subset of ∀ A, ( A∩ ⊆ A) ( A∩ ⊆ )
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!enn Dia"ram Jepresentation for
Intersection
BA
A ∩ B
35(
2
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1 people were surveyed. 8# people in a survey owned acat. $L people owned a do". #7 did not own a do" or cat.
Draw a !enn dia"ram.
universal set is 1 people surveyed
C D
Set ! is t'e cat owners and Set " is t'e do"
owners. &'e sets are N/& dis0oint. Somepeople could own bot' a do" and a cat.
24
Since #7did not owna do" orcat, t'eremust be KLt'at do.
n)C ∪ D* + (6
&'is n means t'enumber of elementsin t'e set
8# W $L * 66 sot'ere must be66 2 KL * 1#people t'at own
bot' a do" anda cat.
12
40 24
Coun%in! o"mula:
n) A
∪ B
* +n) A
* -n) B
* .n) A
∩ B
*
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Set /perations 2 Difference
&'e difference or t'e relative complement of aset P<M wit' respect to a set PM is t'e set ofelements w'ic' belon" to PM but w'ic' do not
belon" to P<M. &'is is denoted by <. In ot'er words we can say
< * @F F∈ ∧ F∉<A
+.". * @$, 8, KA, < * @#, $, 8A < * @$, 8, KA @#, $, 8A * @KA
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Set Difference
=or sets A, , t'e difference of A and ,written A−, is t'e set of all elements t'atare in A but not .
A − ≡ { x | x ∈ A ∧ F∉} = { x | ¬( x ∈ A → x ∈ ) }
lso called&'e com$lement of with res$ect to A.
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!enn Dia"ram Jepresentation for
Difference
BA
A B
35(
2
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Set 5omplements
&'e uni%erse of discourse can itself beconsidered a set, call it U .
&'e com$lement of A, written , is t'ecomplement of A w.r.t. U , i.e., it is U − A.
&.g., If U *N,
A
,}(,6,4,2,1,7{}5,3{ =
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-ore on Set 5omplements
n euivalent definition, w'en U is clear
}8{ A x x A ∉=
AU
A
5 t i 3 d t
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5artesian 3roduct&'e ordered n2tuple (a1, a#, a$, , an) is an orderedcollection of ob0ects.
&wo ordered n2tuples (a1, a#, a$, , an) and
(b1, b#, b$, , bn) are eual if and only if t'ey contain eFactly
t'e same elements in t'e same order , i.e. ai * bi for 1 ≤ i ≤
n.&'e 5artesian product of two sets is defined as
×< * @(a, b) B a∈ ∧ b∈<A
+Fample * @F, yA, < * @a, b, cA ×< * @(F, a), (F, b), (F, c), (y, a), (y, b), (y, c)A
5 t i 3 d t
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5artesian 3roduct
&'e 5artesian product of two sets is defined as ×< * @(a,
b) B a∈ ∧ b∈<A+Fample
* @"ood, badA, < * @student, profA
×< * @
(good, student),(good, student), ((good, prof),good, prof), (bad, student),(bad, student), (bad, prof)(bad, prof)}}
(student, good),(student, good), (prof, good),(prof, good), (student, bad),(student, bad), (prof, bad)(prof, bad)}} BB××A = {A = {
5 t i 3 d t
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5artesian 3roduct
Note t'at
×∅ * ∅ ∅× * ∅
=or non2empty sets and < ≠< ⇔ ×< ≠ <×
B×<B * BB⋅B<B
&'e 5artesian product of two or more sets is defined as
1× #×× n * @(a1, a#, , an) B ai∈ for 1 ≤ i ≤ nA
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Set Identities
Identity A∪∅* A A∩U * A
Domination A∪U'U A∩∅*∅
Idempotent A∪ A * A ' A∩ A Double complement
5ommutative A∪'∪ A A∩'∩ A
ssociative A∪(∪! )*( A∪)∪! A∩(∩! )*( A∩)∩!
A A =*)
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Some 3roperties
⊆ ∪< and < ⊆ ∪< ∩< ⊆ and ∩< ⊆ <
B∪<B * BB W B<B 2 B∩<B ⊆< ⇒ <c⊆ c
< * ∩<c
If ∩< * Φ t'en we say PM and P<M aredis0oint.
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l"ebra of Sets
Idempotent laws ? ∪ *
? ∩ *
ssociative laws ? ( ∪ <) ∪ 5 * ∪ (< ∪ 5)
? ( ∩ <) ∩ 5 * ∩ (< ∩ 5)
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l"ebra of Sets ctd
5ommutative laws ? ∪ < * < ∪
? ∩ < * < ∩
Distributive laws ? ∪ (< ∩ 5) * ( ∪ <) ∩ ( ∪ 5)
? ∩ (< ∪ 5) * ( ∩ <) ∪ ( ∩ 5)
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l"ebra of Sets ctd
Identity laws ? ∪ Φ *
? ∩ *
? ∪ * ? ∩ Φ * Φ
Involution laws ? (c)c *
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l"ebra of Sets ctd
5omplement laws ? ∪ c *
? ∩ c * Φ
? c * Φ Φc *
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l"ebra of Sets ctd
De -or"anMs laws ? ( ∪ <)c * c ∩ <c
? ( ∩ <)c * c ∪ <c
Note 5ompare t'ese De -or"anMs lawswit' t'e De -or"anMs laws t'at you find inlo"ic and see t'e similarity.
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3rovin" Set Identities
&o prove statements about sets, of t'e form& 1 * & # (w'ere & s are set eFpressions),'ere are t'ree useful tec'niues
3rove & 1 ⊆ & # and & # ⊆ & 1 separately. se lo"ical euivalences.
se a membershi$ table.
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3roofs
<asically t'ere are two approac'es inprovin" above mentioned laws and anyot'er set relations'ip
? l"ebraic met'od ? sin" !enn dia"rams
=or eFample lets discuss 'ow to prove
? ( ∪ <)c * c ∩ <c
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3roofs sin" l"ebraic -et'od
F∈(∪<)c ⇒ F∉ ∪<⇒ F∉ ∧ F∉<
⇒ F∈ c
∧ F∈
<c
⇒ F∈ c∩<c
⇒ (∪<)c ⊆ c∩<c
)α*
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3roofs sin" l"ebraic -et'od
ctdF∈ c∩<c ⇒ F∈ c ∧ F∈<c
⇒ F∉ ∧ F∉<
⇒ F∉ ∪<
⇒ F∈(∪<)c
⇒ c
∩<c
⊆ (∪<)c
)β*
)α* ∧ )β* ⇒ )A∪B*' + A'∩B'
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3roofs sin" !enn Dia"rams
Note t'at t'ese indicated numbers arenot t'e actual members of eac' set.&'ey are re"ion numbers.
BA
A ∪ B
3
41
2
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3roofs sin" !enn Dia"rams ctd
1, #, $, 7
1, # (i.e. &'e re"ion for PM is 1 and #)
< #, $∴ ∪< 1, #, $
∴ (∪<)c 7 )α*
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3roofs sin" !enn Dia"rams ctd
c $, 7<c 1, 7
∴ c∩<c 7)β*
)α* ∧ )β* ⇒ )A∪B*' + A'∩B'
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5lass +Fercise 2 1
Let A !a, b, c, d", # !a, b, c" and$ !b, d". %ind all sets & suc' t'at
(i) (ii)
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Solution(i)
( ) { } { } { } { } { } { } { }{ }= φ3 < , a , b , c , a, b , a, c , b, c , a, b, c
( ) { } { } { }{ }= φ3 5 , b , d , b, d
⊂ ⊂X < and X 5Q
( ) ( ) ( ) ( )⇒ ∈ ∈ ⇒ ∈X 3 < and X 3 5 X 3 < 3 5I
{ }⇒ = φX , b
⊂ ⊄(ii) Now, X and X <
& is subset of A but & is not subset of #.⇒( ) ( ) ( ) ( )⇒ ∈ ∉ ⇒ ∈ −X 3 but F 3 < X 3 3 <
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Solution contd..
∴ =X !d", !a, d" !b, d", !c, d", !a, b, d", !a, c, d",
!b, c, d", !a, b, c, d"
ere note t'at to obtain & we 'ave added eac'element of (#) wit' *d’ w'ic' is in A not in #.
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5lass +Fercise 2 #
%or any two sets A and #, prove t'at
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Solution%irst let A #. +'en
= = < and < U I
⇒ = < <U I
∴ = ⇒ = < < < ...(i)U I
=5onversely, let < <.U I
∴ ∈ ⇒ ∈F F <U
⇒ ∈F <I
⇒ ∈ ∈F and F <
⇒ ∈F <
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Solution contd..∴ ⊆ < ...(ii)
ow let∈ ⇒ ∈y < y <U
⇒ ∈y <I
⇒ ∈ ∈y and y <
⇒ ∈y
∴ ⊆< ...(iii)
%rom (ii) and (iii), we get A #
= ⇒ =&'us, < < < ...(iv)U I
= ⇔ ==rom (i) and (iv), < < <U I
5l + i $
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5lass +Fercise 2 $
f suc' t'at ,describe t'e set
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Solutione 'ave { }= ∈aN aFF N
{ } { }∴ = ∈ =$N $FF N $, L, 9, 1#, 18, ...
{ } { }= ∈ =KN KFF N K, 17, #1, #6, ...
{ }=Yence, $N KN #1, 7#, L$, ...I { }= ∈ =#1FF N #1N
ote t'at w'ere c L$M of a, b.=aN bN cNI
5l + i 7
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5lass +Fercise 2 7
f A, # and $ are any t'ree sets, t'enprove t'at
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SolutionLet / be any element of .( )− < 5I
( ) ( )∴ ∈ − ⇒ ∈ ∉F < 5 F and F < 5I I
( )⇒ ∈ ∉ ∉F and F < or F 5
( ) ( )⇒ ∈ ∉ ∈ ∉F and F < or F and F 5
( ) ( )⇒ ∈ − ∈ −F < or F 5
( ) ( )⇒ ∈ − −F < 5U
( ) ( ) ( )∴ − ⊆ − − < 5 < 5 ...(i)I U
Solution contd
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Solution contd..Again y be any element of .( ) ( )− − < 5U
( ) ( ) ( ) ( )∴ ∈ − − ⇒ ∈ − ∈ −y < 5 y < or y 5U
( ) ( )⇒ ∈ ∉ ∈ ∉y and y < or y and y 5
( )⇒ ∈ ∉ ∉y and y < or y 5
( )( )⇒ ∈ ∉y and y < 5I ( )⇒ ∈ −y < 5I
( ) ( ) ( )∴ − − ⊆ − < 5 < 5 ...(ii)U I
%rom (i) and (ii),
( ) ( ) ( )− = − − < 5 < 5 3r oved.I U
5l + i 8
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5lass +Fercise 2 8
Let A, # and $ be t'ree sets suc' t'at and , t'en prove t'atA $ 0 #
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SolutionQ =>e 'ave < 5.U
( )∴ − = −5 < < <U
( ) [ ]′ ′= − = < < X Z X ZU I Q I
( ) ( )′ ′= < < < [<y distributive law\I U I
( )′= φ <I U
′= <I
* ? <
[ ]= φ* < 3r oved.Q I
5l + i L
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5lass +Fercise 2 L
f A, # and $ are t'e sets suc' t'at t'en prove t'at
Solution
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SolutionLet / be any arbitrary element of $ 0 #.
∴ ∈ − ⇒ ∈ ∉F 5 < F 5 and F <[ ]⇒ ∈ ∉ ⊂F 5 and F <Q
⇒ ∈ −F 5
∴ − ⊂ −5 < 5 3r oved.
5l + i K
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5lass +Fercise 2 K
f A, # and $ are t'e t'ree sets and 1is t'e universal set suc' t'at n(1) 233, n(A) 433, n(#) 533 and, find
Solution
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( ) ′′ ′ = < <Q I U #y 6e Morgan’s law
( ) ( )( )′′ ′∴ =n < n <I U
( ) ( )= −n n <U
( ) ( ) ( ) ( ) = − + − n n n < n <I
233 0 7433 8 533 0 933:
533
5l + i 6
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5lass +Fercise 2 6
n a class of 5; students, 92 'aveta<en mat'ematics, 93 'ave ta<enmat'ematics but not p'ysics. %ind
t'e number of students w'o 'aveta<en bot' mat'ematics and p'ysicsand t'e number of students w'o 'aveta<en p'ysics but not mat'ematics,if it is given t'at eac' student 'as
ta<en eit'er mat'ematics or p'ysicsor bot'.
S l ti
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SolutionMethod I:
Let M denote t'e set of students w'o'ave ta<en mat'ematics and be t'eset of students w'o 'ave ta<en p'ysics.
( ) ( ) ( )= = − =n - 3 $8, n - 1K, n - 3 1DU
=iven t'at
( ) ( ) ( )=>e ;now t'at n - ? 3 n - ? n - 3I
( )⇒ = −1 1K n - 3I
( )⇒ = − = ⇒n - 3 1K 1 KI 2 students 'ave ta<enbot' mat'ematics and p'ysics.
Solution contd..
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Solution contd..ow we want to find n( 0 M).
( ) ( ) ( ) ( )∴ = + −n - 3 n - n 3 n - 3U I
⇒ 5; 92 8 n() 0 2
⇒ n() 5; 0 93 4;
( ) ( ) ( )∴ − = −n 3 - n 3 3 - 3I
4; 0 2 9>
⇒ 9> students 'ave ta<en p'ysics but not mat'ematics.
S l ti td
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Solution contd..Method II:
Venn diagram met'od?
- 3
a b c
( ) = + + =4iven t'at n - 3 a b c $8 ...(i)U
n(M) a 8 b 92 ...(ii)
n(M 0 ) a 93 ...(iii)
S l ti td
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Solution contd..e want to find b and c
%rom (ii) and (iii),b 92 0 93 2 2 students 'aveta<en bot' p'ysics and mat'ematics.
⇒
%rom (i), 93 8 2 8 c 5;
c 5; 0 92 9>
⇒ 9> students 'ave ta<en p'ysics but not mat'ematics.
5lass +Fercise 9
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5lass +Fercise 2 9
f A and # be t'e two sets containing5 and @ elements respectively, w'at
can be t'e minimum and ma/imumnumber of elements in
S l ti
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SolutionAs we <now t'at,
( ) ( ) ( ) ( )= + −n < n n < n <U I
( )∴ n <U is minimum or ma/imum accordingly as
( )n <I is ma/imum or minimum respectively.
Case I: 'en is minimum, i.e. 3( )n <I ( )n <I
+'is is possible only w'en .= φ <I
( ) ( ) ( ) ( )∴ = + −n < n n < n <U I
5 8 @ 0 3
∴ Ma/imum number of elements in <U
Solution contd..
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Case II: 'en is ma/imum( )n <I
+'is is possible only w'en . n t'is case⊆ <
= < <U
( ) ( )∴ = =n < n < LU
∴ Minimum number of elements in is @. <U
5lass +Fercise 1
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5lass +Fercise 2 1
But of >>3 boys in a sc'ool, 44C playcric<et, 4C3 play 'oc<ey, and 55@ playbas<etball. Bf t'e total, @C play bot'
bas<etball and 'oc<eyD >3 play cric<etand bas<etball and C3 play cric<et and'oc<eyD 4C play all t'e t'ree games.%ind t'e number of boys w'o did notplay any game.
S l ti
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SolutionMethod I:
Let $, # and denote t'e set of boysplaying cric<et, bas<etball and 'oc<eyrespectively.
ere given t'at
n($) 44C, n() 4C3, n(#) 55@
( ) ( ) ( )= = =n < Y L7, n 5 < 6, n 5 Y 7I I I
( ) =n 5 < Y #7I I
Q e <now t'at
( ) ( ) ( ) ( ) ( )= + + − −n 5 < Y n 5 n < n Y n 5 <U U I
( ) ( ) ( )− +n < Y n 5 Y n 5 < YI I I I
44C 8 55@ 8 4C3 0 >3 0 @C 0 C3 8 4C
@C3
Solution contd
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Solution contd..∴ umber of boys not playing any game is
+otal number of students 0 ( )n 5 < YU U
>>3 0 @C3 4C3
Method II:
Venn diagram met'od?
a b c
d
e
f
"
5 <
Y
t is given t'at
n($) a 8 b 8 d 8 e 44C ...(i)n() d 8 e 8 f 8 g 4C3 ...(ii)
n(#) b 8 c 8 e 8 f 55@ ...(iii)
Solution contd..
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( )n < Y * e W f * L7 ...(iv)I
( )n 5 < * b W e * 6D ...(v)I
( )n 5 Y * d W e * 7D ...(vi)I
( )n 5 < Y * e * #7 ...(vii)I I
+ = ⇒ = − =d e 7D d 7D #7 1LQ
+ = ⇒ = − =b e 6D b 6D #7 8L
+ = ⇒ = − =e f L7 f L7 #7 7D
+ + + = ⇒ = − − −b c e f $$L c $$L 8L #7 7Q 49@
"ain d W e W f W " * #7 " * #7 ? 1L ? #7 ? 7
* #7 ? 6
* 1L
⇒
Solution contd..
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and a 8 b 8 d 8 e 44C
a 44C 0 ;@ 0 9@ 0 4C 44C 0 @
94>
⇒
∴ EeFuired number of students not playing any game >>3 0 (a 8 b 8 c 8 d 8 e 8 f 8 g)
>>3 0 (94> 8 ;@ 8 49@ 8 9@ 8 4C 8 C3 8 9@3)
>>3 0 @C3
4C3
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