bm cycle 7 session 1

7/23/2019 BM Cycle 7 Session 1

http://slidepdf.com/reader/full/bm-cycle-7-session-1 1/95

PAN African e-Network Project

PGD IT

BASIC MATHEMATICS

Semester - I

Session - 1

Dr. Nitin Pane!



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



Instructional /b0ectives

Illustrate properties of set al"ebra usin"!enn2dia"rams.

3rove various useful results of set al"ebra.



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



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.



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.



<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



+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)



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.



<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



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.



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



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 ∈ )



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



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



&'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 ∈∅.



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



!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



<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”



+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



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.



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 ∈ <)



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



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 ?



Subsets seful rules

* < ⇔ ( <) (< ) ( <) (< 5) ⇒ 5 (see !enn Dia"ram)

UU

AABB

CC



Some 3roperties Je"ardin"

Subsets =or any set PM, Φ ⊆ ⊆

=or any set PM, ⊆

⊆ < ∧ < ⊆ 5 ⇒ ⊆ 5

* < ⇔ ⊆ < ∧ < ⊆



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∈< ∧ ≠<)



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



!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



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.


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.



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



!enn Dia"ram for t'e 5omplement

A

A'



&'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∪ ⊇ )



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 ∪ <.


∪ < * @F F∈ ∨ F∈<A

+.". * @$, 8, KA, < * @#, $, 8A ∪ < * @$, 8, K, #, $, 8A * @#, $, 8, KA



@a,b,cA∪@#,$A * @a,b,c,#,$A

@#,$,8A∪@$,8,KA * @#,$,8,$,8,KA *@#,$,8,KA

nion +Famples



!enn Dia"ram Jepresentation for

nion

BA

A ∪ B

35(

2



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 ∩ <.


∩ < * @F F∈ ∧ F∈<A

+.". * @$, 8, KA, < * @#, $, 8A ∩ < * @$, 8A



&'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∩ ⊆ )




Intersection

BA

A ∩ B

35(

2



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

*



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



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.




Difference

BA

A B

35(

2



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{ =



-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



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



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



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



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 =*)



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.



l"ebra of Sets

Idempotent laws ? ∪ *

? ∩ *

ssociative laws ? ( ∪ <) ∪ 5 * ∪ (< ∪ 5)

? ( ∩ <) ∩ 5 * ∩ (< ∩ 5)



l"ebra of Sets ctd

5ommutative laws ? ∪ < * < ∪

? ∩ < * < ∩

Distributive laws ? ∪ (< ∩ 5) * ( ∪ <) ∩ ( ∪ 5)

? ∩ (< ∪ 5) * ( ∩ <) ∪ ( ∩ 5)



l"ebra of Sets ctd

Identity laws ? ∪ Φ *

? ∩ *

? ∪ * ? ∩ Φ * Φ

Involution laws ? (c)c *



l"ebra of Sets ctd

5omplement laws ? ∪ c *

? ∩ c * Φ

? c * Φ Φc *



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.



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.



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



3roofs sin" l"ebraic -et'od

F∈(∪<)c ⇒ F∉ ∪<⇒ F∉ ∧ F∉<

⇒ F∈ c

∧ F∈

<c

⇒ F∈ c∩<c

⇒ (∪<)c ⊆ c∩<c

)α*



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'



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



3roofs sin" !enn Dia"rams ctd

1, #, $, 7

1, # (i.e. &'e re"ion for PM is 1 and #)

< #, $∴ ∪< 1, #, $

∴ (∪<)c 7 )α*



3roofs sin" !enn Dia"rams ctd

c $, 7<c 1, 7

∴ c∩<c 7)β*

)α* ∧ )β* ⇒ )A∪B*' + A'∩B'



5lass +Fercise 2 1

Let A !a, b, c, d", # !a, b, c" and$ !b, d". %ind all sets & suc' t'at

(i) (ii)



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 <



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 #.



5lass +Fercise 2 #

%or any two sets A and #, prove t'at



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 <



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 $



5lass +Fercise 2 $

f suc' t'at ,describe t'e set



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



5lass +Fercise 2 7

f A, # and $ are any t'ree sets, t'enprove t'at



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



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



5lass +Fercise 2 8

Let A, # and $ be t'ree sets suc' t'at and , t'en prove t'atA $ 0 #



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



5lass +Fercise 2 L

f A, # and $ are t'e sets suc' t'at t'en prove t'at

Solution



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



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



( ) ′′ ′ = < <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



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



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..



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



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



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



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



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..



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



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



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



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..



( )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..



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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