Sets. Sets. Operations on Sets Laws of Algebra of Sets Cardinal Number of a Finite and Infinite Set. Representation of Sets Power Set Venn Diagram

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1 Sets MILESTONE Sets Represetatio of Sets Power Set Ve Diagram Operatios o Sets Laws of lgebra of Sets ardial Number of a Fiite ad Ifiite Set I Mathematical laguage all livig ad o-livig thigs i uiverse are kow as objects. The collectio of well defied distict objects is kow as a set. Well defied meas i a give set, it must be possible to decide whether or ot the object belogs to the set ad by distict meas object should ot be repeated. The object i the set is called its member or elemet. set is represeted by { }. Geerally, sets are deoted by capital letters,,,... ad its elemets are deoted by small letters a, b, c,..... Let is a o-empty set. If x is a elemet of, the we write x ad read as x is a elemet of or x belogs to. If x is ot a elemet of, the we write x ad read as x is ot a elemet of or x does ot belog to. e.g., Set of all vowels i Eglish alphabets. I this set a, e, i, o adu are members. The theory of sets was developed by Germa Mathematicia Georg ator ( ). The cocept of sets is widely used i the foudatio of relatios, fuctios, logic, probability theory, etc. ccordig to ator set is ay collectio ito a whole of defiite ad distict objects of our ituitio or thought. Sample Problem 1 Which of the followig is a correct set? (a) The collectio of all the moths of a year begiig with the letter J. (b) The collectio of te most taleted writers of Idia. (c) team of eleve best cricket batsme of the world. (d) collectio of most dagerous aimals of the world. Iterpret (a) (a) We are sure that members of this collectio are Jauary, Jue ad July. So, this collectio is well defied ad hece, it is a set. (b) writer may be most taleted for oe perso ad may ot be for other. Therefore, it caot be said accoutely that which writers will be there i the relatio. So, this collectio is ot well defied. Hece, it is ot a set.

2 4 JEE Mai Mathematics (c) batsma may be best for oe perso ad may ot be so for other. Therefore, it caot be said accoutely that which batsma will be there i our relatio. So, this collectio is ot well defied. Hece, it is ot a set. (d) The term most dagerous is ot a clear term. aimal may be most dagerous to oe perso ad may ot be for the other. So, it is ot well defied, hece it is ot a set. We ca use the followig two methods to represet a set. (i) Listig Method I this method, elemets are listed ad put withi a braces { } ad separated by commas. This method is also kow as Tabular method or Roster method. e.g., Set of all prime umbers less tha 11 = {, 3, 5, 7} (ii) Set uilder Method I this method, istead of listig all elemets of a set, we list the property or properties satisfied by the elemets of set ad write it as = { x : P( x)} or { x P( x)} It is read as is the set of all elemets x such that x has the property P( x). The symbol : or stads for such that. This method is also kow as Rule method or Property method. e.g., = { 1,, 3, 4, 5, 6, 7, 8} = { x : x N ad x 8} The order of elemets i a set has o importace e.g., {1,, 3} ad {3, 1, } are same sets. The repetitio of elemets i a set does ot effect the set, e.g., {1,, 3} ad {1, 1,, 3} both are same sets. (i) Set of all atural umbers, N { 1,, 3,...} (ii) Set of all whole umbers, W { 0, 1,, 3,...} (iii) (a) Set of all itegers, I or Z {...,, 1, 0, 1,,...} (b) Set of all positive or egative itegers, I { 1,, 3,... } or I { 1,, 3,... } (c) Set of all eve (E) or odd (O) itegers, E {..., 4,, 0,, 4,...} or O {..., 3, 1, 0, 1, 3,...} (iv) (a) Set of all ratioal umbers, Q {p/ q, where pad q are itegers ad q 0} (b) Set of all irratioal umbers, IR {which caot be p ad I I, q 0} (c) Set of all real umbers, R { x : x } (v) Set of all complex umbers, { a ib; a, b R ad i 1} Sample Problem The builder form of followig set is { 3, 6, 9, 1 }, {, 1 4, 9,..., 100} (a) { x : x 3, N 5}, { x : x, N 10} (b) { x : x 3, N 4}, { x : x, N 10} (c) { x : x 3, N 4}, { x : x, N 10} (d) Noe of the above Iterpret (b) Give, { 3, 6, 9, 1} ad { x : x 3, N 4} ad { 1, 4, 9,..., 100} { x : x, N 10 } set which has o elemet, is called a empty set. It is deoted by or { }. e.g., Set of all odd umbers divisible by ad = {x : x N ad 5 x 6} Such sets which have atleast oe elemet, are called o-void set. If represets a ull set, the is ever writte with i braces i.e., { } is ot the ull set. set which have oly oe elemet, is called a sigleto set. e.g., = { x : x N ad 3 x 5} ad = { 5} set i which the process of coutig of elemets surely comes to a ed, is called a fiite set. I other words set havig fiite umber of elemets is called a fiite set.

3 Sets 5 Otherwise it is called ifiite set i.e., if the process of coutig of elemets does ot come to a ed i a set, the set is called a ifiite set. e.g., { x : x N ad x 5} ad Set of all poits o a plae I above two sets ad, set is fiite while set is ifiite. Sice, i a plae ay umber of poits are possible. ad read as is ot a subset of. e.g., If = {1,, 3} ad = {1,, 3, 4, 5} Here, each elemet of is a elemet of. Thus, i.e., is a subset of ad is a superset of. Null set is a subset of each set. Each set is a subset of itself. If has elemets, the umber of subsets of set is. Two fiite sets ad are said to be equivalet, if they have the same umber of elemets. e.g., If { 1,, 3} ad { 3, 7, 9} Number of elemets i 3 adumber of elemets i 3 ad are equivalet sets. If ad are two o-empty sets ad each elemet of set is a elemet of set ad each elemet of set is a elemet of set, the sets ad are called equal sets. Symbolically, if x x ad x x e.g., = {1,, 3} ad = {x : x N, x 3} Here, each elemet of is a elemet of, also each elemet of is a elemet of, the both sets are called equal sets. Equal sets are equivalet sets while its coverse eed ot to be true. If each elemet of is i set but set has atleast oe elemet which is ot i, the set is kow as proper subset of set. If is a proper subset of, the it is writte as ad read as is a proper subset of. e.g., If ad N= {1,, 3, 4, } I = {, 3,, 1, 0, 1,, 3, } the N I If has elemets, the umber of proper subsets is 1. Two sets ad are said to be comparable, if either or or, otherwise, ad are said to be icomparable. e.g., Suppose { 1,, 3}, { 1,, 4, 6} ad { 1,, 4} Sice, or or ad are icomparable. ut ad are comparable sets. Let ad be two o-empty sets. If each elemet of set is a elemet of set, the set is kow as subset of set. If set is a subset of set, the set is called the superset of. lso, if is a subset of, the it is deoted as ad read as is a subset of. Thus, if x x, the If x x, the If there are some sets uder cosideratio, the there happes to be a set which is a superset of each oe of the give sets. Such a set is kow as the uiversal set ad it is deoted by S or. This set ca be chose arbitrarily for ay discussio of give sets but after choosig it is fixed. e.g., Suppose = {1,, 3}, = {3, 4, 5} ad = {7, 8, 9} = {1,, 3, 4, 5, 6, 7, 8, 9} is uiversal set for all three sets. Sample Problem 3 osider the followig sets The set of lies which are parallel to the X-axis. The set of letters i the Eglish alphabet. ad The set of aimals livig o the earth. Which of these is fiite or ifiite set? (a) Fiite set,, Ifiite set (b) Fiite set,, Ifiite set (c) Fiite set,, Ifiite set (d) Noe of the above Iterpret (b) Ifiite lies ca be draw parallel to X-axis There are fiite 6 Eglish alphabets There are fiite umber of aimals livig o earth

4 6 JEE Mai Mathematics Sample Problem 4 Two fiite sets have m ad elemets, respectively. The total umber of subsets of the first set is 56 more tha the total umber of subsets of secod set. What are the values of m ad, respectively? (a) 7, 6 (b) 6, 3 (c) 5, 1 (d) 8, 7 Iterpret (b) Sice, total possible subsets of sets ad are m ad, respectively. ccordig to give coditio, m 56 m 3 3 ( 1) ( 1) O comparig both sides, we get 3 ad m 3 3 ad m 3 m 6 ad 3 Let be a o-empty set, the collectio of all possible subsets of set is kow as power set. It is deoted by P( ). e.g., Suppose = {1,, 3} P( ) = [, {1}, {}, {3}, {1, }, {, 3}, {3, 1}, {1,, 3}]. (a) P( ) (b) { } P( ) { 1,, 3} P( ) {, {}, 1 { }, { 3}, { 1, }, {, 3}, { 1, 3}, { 1,, 3}} P( ) P( ) {, {}, 1 { }, { 3}, {, 1 }, {, 3}} P( ) P( ) P( ) Swiss Mathematicia ( ) Euler gave a idea to represet a set by the poits i a closed curve. Later o ritish Mathematicia Joh Ve ( ) brought this idea to practice. So, the diagrams draw to represet sets are called Ve Euler diagram or simply Ve diagram. I Ve diagram, the uiversal set is represeted by a rectagular regio ad a set is represeted by circle or a closed geometrical figure iside the uiversal set. lso, a elemet of a set is represeted by a poit withi the circle of set. e.g., If = {1,, 3, 4,, 10} ad = {1,, 3} The, its Ve diagram is as show i the figure (i) Each elemet of a power set is a set. (ii) If, the P( ) P( ) (iii) Power set of ay set is always o-empty. (iv) If set has elemets, the P( ) has elemets. (v) P( ) P( ) P( ) (vi) P( ) ( ) P( ) (vii) P( ) P( ) P( ) Sample Problem 5 If set {, 1 3, 5 }, the umber of elemets i P{ P( )} is (a) 8 (b) 56 (c) 48 (d) 50 Iterpret (b) Give, { 1, 3, 5} 3 { P( )} 8 [ P{ P( )}] 56 Sample Problem 6 osider { 1, }, {, 3 }. The which of the followig optio is correct? (a) P( ) P( ) P( ) (b) P( ) P( ) P( ) (c) P( ) P( ) P( ) (d) Noe of these Iterpret (a) Here P( ) {, { 1}, { }, { 1, }}, P( ) {,{ }, { 3}, {, 3}} 8 Now, we itroduce some operatios o sets to costruct ew sets from the give oes. Let ad be two sets, the uio of ad is a set of all those elemets which are i or i or i both ad. It is deoted by ad read as uio. Symbolically, = {x : x or x } learly, x x or x If x x ad x The ve diagram of is as show i the figure ad the shaded portio represets. (whe ) whe either or e.g., If = {1,, 3, 4} ad = {4, 8, 5, 6} = {1,, 3, 4, 5, 6, 8}. whe ad are disjoit sets

5 Sets 7 The uio of a fiite umber of sets 1,,..., is represeted by i i or Symbolically, { x : x for atleast oe i} i 1 i If ad are two sets, the itersectio of ad is a set of all those elemets which are i both ad. The itersectio of ad is deoted by ad read as itersectio. Symbolically, = {x : x ad x } If x x ad x ad if x x or x The Ve diagram of is as show i the figure ad the shaded regio represets. i If ad are two o-empty sets, the differece of ad is a set of all those elemets which are i but ot i. It is deoted as. If differece of two sets is, the it is a set of those elemets which are i but ot i. Hece, = {x : x ad x } ad = {x : x ad x } If x x but x ad if x x but x The Ve diagram of ad are as show i the figure ad shaded regio represets ad. whe, i.e., ( = φ) whe whe either whe or = or e.g., If = {1,, 3, 4} ad = {4, 3, 5, 6} = {3, 4} The itersectio of a fiite umber of sets 1,, 3,..., is represeted by i i or Symbolically, { x : x for all i} i 1 i Two sets ad are kow as disjoit sets, if i.e., if ad have o commo elemet. The Ve diagram of disjoit sets as show i the figure i e.g., If = {1,, 3} ad = {4, 5, 6}, the { } ad are disjoit sets. = φ (o shaded regio) e.g., If = {1,, 3, 4} ad = {4, 5, 6, 7, 8} = {1,, 3} ad = {5, 6, 7, 8} ad ad The sets ad are disjoit sets. If ad are two sets, the set ( ) ( ) is kow as symmetric differece of sets ad ad is deoted by. The Ve diagram of is as show i the figure ad shaded regio represets. e.g., = {1,, 3} ad = {3, 4, 5, 6}, the = ( ) ( ) = {1, } {4, 5, 6} = {1,, 4, 5, 6} whe either or Symmetric differece ca also be writte as (commutative) ( ) ( ) whe ad are disjoit sets. learly, =

6 8 JEE Mai Mathematics The complemet of a set is the set of all those elemets which are i uiversal set but ot i. It is deoted by or c. If is a uiversal set ad, the = {x : x but x } i.e., x x The Ve diagram of complemet of a set is as show i the figure ad shaded portio represets. e.g., If ad = {1,, 3, 4, 5, } = {, 4, 6, 8, } = { 1, 3, 5, 7, } y Ve diagram, the operatio betwee three sets ca be represeted give below. ( ) Sample Problem 7 If {, 1, 3, 4, 5, 6, 7, 8, 9 }, {, 4, 6, 8} ad {, 3, 5, 7 }, the ( ), ( ), ( ) is equal to (a) {1, 9}, {, 8}, {3, 4, 5, 6, 7, 8} (b) {1, 9}, {1, 9} {3, 4, 5, 6, 7, 8} (c) {1, 9}, {1, 9} {5, 6, 7, 8} (d) Noe of the above Iterpret (b) Give sets are { 1,, 3, 4, 5, 6, 7, 8, 9 }, {, 4, 6, 8} ad {, 3, 5, 7} Now, ad { 1,, 3,, 4, 5, 6, 7, 8, 9} {, 4, 6, 8} { 1, 3, 5, 7, 9} { 1,, 3, 4, 5, 6, 7, 8, 9} {, 3, 5, 7} { 1, 4, 6, 8, 9} ' (i) {, 4, 6, 8} {, 3, 5, 7} {, 3, 4, 5, 6, 7, 8} ( ) { 1,, 3, 4, 5, 6, 7, 8, 9} {, 3, 4, 5, 6, 7, 8} { 1, 9} (ii) ( ) { 1, 3, 5, 7, 9} { 1, 4, 6, 8, 9} { 1, 9} (iii) Now, {, 4, 6, 8} {, 3, 5, 7} { 4, 6, 8} ad {, 3, 5, 7} {, 4, 6, 8} { 3, 5, 7} ( ) ( ) { 4, 6, 8} { 3, 5, 7} { 3, 4, 5, 6, 7, 8} Sample Problem 8 The shaded regio i the give figure is (a) ( ) (b) ( ) (c) ( ) (d) ( ) Iterpret (d) It is clear from the figure that set is ot shadig ad set is shadig other tha. i.e., ( ) Sample Problem 9 If {x : x 6x 11x 6x 0}, {x : x 5x 6 0} ad {x : x 3x 0} what is ( ) equal to? (a) {1, 3} (b) {1,, 3} (c) {0, 1, 3} (d) {0, 1,, 3} Iterpret (c)q {x : x 6x 11x 6x 0} {0, 1,, 3} {x : x 5x 6 0} {, 3} ad { x : x 3x 0} { 1, } { } Hece, ( ) ( ) {0, 1,, 3} {} {0, 1, 3} If, ad are three o-empty sets, the (i) Idempotet law (a) (b) (ii) Idetity law (a) (b)

7 Sets 9 (iii) ommutative law (a) (b) (iv) ssociative law (a) ( ) ( ) (b) ( ) ( ) (v) Distributive law (a) ( ) ( ) ( ) (b) ( ) ( ) ( ) (vi) De-morga s law (a) ( ) (b) ( ) (c) ( ) ( ) ( ) (d) ( ) ( ) ( ) Sample Problem 10 If ad are two sets, the ( ) is equal to (a) (b) (c) (d) Noe of these Iterpret (c) ( ) ( ) (Qby de-morga s law) ( ) (Q by associative law) (Q ) Sample Problem 11 If ad are o-empty sets, the ( ) ( ) is equal to (a) (b) (c) (d) Iterpret (b) ( ) ( ) ( ) ( ) [ Q( ) ( )] ( ) (Q by distributive law) (Q ) (iv) ( ) ( ) ( ) (v) ( ( ) ( ) ( ) (vi) ( ) ( ) ( ) (vii) ( ) ( ) ( ) (viii) ( ) ( ) ( ) (ix) ( ) ( ) ( ) Method to Fid ommo Roots Sometimes the umber of commo elemets caot foud easily y by solvig the give sets. That type of problems ca be solved by drawig a curves. The itersectio poit of a curve is equal to the umber of commo elemets i a set. e.g., osider the sets 1 {( x, y) y, 0 x R} x ad {( x, y) y x, x R}, the determie ( ). Here, we see that, x, we get ifiite values of y. Hece, we fid ifiite sets ad. d it is difficult to fid the commo elemets betwee ad. Now, firstly we make the graph of give sets. The set of all poits o a curve xy 1, [Qxy c is a rectagular hyperbola curve] ad The set of all poits o a curve y x. [Qy x is a straight which passes through origi] x' y xy = 1 x y = x y' Sice, there is o itersectio poit o a curve. So, there is o commo elemets betwee two sets. The umber of distict elemets i a fiite set is called cardial umber ad it is deoted by ( ). d if it is ot fiite set, the it is called ifiite set. e.g., If { 3, 1, 8, 10, 13, 17, } the ( ) 6 If, ad are fiite sets ad be the fiite uiversal set, the (i) ( ) ( ) ( ) ( ) (ii) If ad are disjoit sets the, ( ) ( ) ( ) (iii) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Sample Problem 1 I a tow of families it was foud that 40% families buy ewspaper, 0% families buy ewspaper ad 10% families buy ewspaper, 5% buy ad, 3% buy ad ad 4% buy ad. If % families buy all of three ewspapers, the the umber of families which buy oly, is (a) 4400 (b) 3300 (c) 000 (d) 500 Iterpret (b) ( ) 40% of , ( ) 000, ( ) 1000, ( ) 500, ( ) 300, ( ) 400, ( ) 00 ( ) { ( ) } ( ) { ( )} ( ) ( ) ( ) ( )

8 WORKED OT Example 1 If { 3, 5, 7, 9, 11 }, { 7, 9, 11, 13 }, { 11, 13, 15} add { 15, 17 }, the( D) ( ) is equal to (a) {5, 7, 9, 11, 15} (b) {7, 9, 11, 15} (c) {7, 9, 11, 13, 15} (d) Noe of these Solutio (b) Give, { 3, 5, 7, 9, 11, } { 7, 9, 11, 13 }, { 11, 13, 15} ad D { 15, 17} Now, D { 3, 5, 7, 9, 11} { 15, 17} { 3, 5, 7, 9, 11, 15, 17} ad { 7, 9, 11, 13} { 11, 13, 15} { 7, 9, 11, 13, 15} ( D) ( ) { 3, 5, 7, 9, 11, 15, 17} { 7, 9, 11, 13, 15} { 7, 9, 11, 15} Example The set ( ) ( ) is equal to (a) (b) (c) (d) Noe of these Solutio (a) ( ) ( ) Example 3 Let ( ) ( ) ( ) ( ) ( ), ad are subsets of uiversal set. If {, 4, 6, 8, 1, 0 }, { 3, 6, 9, 1, 15 }, { 5, 10, 15, 0} ad is the set of all whole umbers. The, the correct Ve diagram is ( a) ( c) ( b) Solutio (b) Give, {, 4, 6, 8, 1, 0} { 3, 6, 9, 1, 15} ( d) Noe of these { 5, 10, 15, 0} Now, {, 4, 6, 8, 1, 0} { 3, 6, 9, 1, 15} { 6, 1} { 3, 6, 9, 1, 15} { 5, 10, 15, 0} { 15} { 5, 10, 15, 0} {, 4, 6, 8, 1, 0} { 0} ad Example 4 I a survey of 600 studets i a school, 150 studets were foud to be takig tea ad 5 takig coffee. 100 were takig both tea ad coffee. Fid how may studets were takig either tea or coffee? (a) 310 (b) 30 (c) 37 (d) 35 Solutio (d) Let ad T deote the studets takig coffee ad tea, respectively. Here, ( T) 150, ( ) 5, ( T) 100 sig the idetity ( T) ( T) ( ) ( T), we have ( T) ( T) 75 Give, total umber of studets 600 ( ) We are to fid the umber of studets takig either tea or coffee i.e., ( T). ( T) ( ) ( T) Example 5 If there are three atheletic teams i a school, 1 are i the basketball team, 6 i hockey team ad 9 i the football team. 15 play hockey ad basketball, 15 play hockey ad football, 1 play football ad basketball ad 8 play all the games. The total umber of members is (a) 4 (b) 43 (c) 45 (d) Noe of these Solutio (b) Q( ) 1, ( H) 6, ( F) 9, ( H ) 14, ( H F) 15, ( F ) 1, ( H F) 8 ( H F) ( ) ( H) ( F) ( H) ( H F) ( F) ( H F)