1 Sets, Relations and Functions (i) Listing method In this method, the set is Sets represented by writing elements in a bracket and separatedbycomma(,). In our mathematical language, everything in this = universewhetherlivingornon-livingiscalledanobject. e.g., A SetofvowelsofEnglishalphabet Any collection of well defined objects is called a set. By = {a , e , i , o , u }. ‘well defined objects’, we mean that given a set and an This method is also known as Tabular method or object, it must be possible to decide whether or not the Rostermethod. object belongs to the set. The objects in set are called its (ii) Set builder method In this method, instead membersorelements. of listing all elements of a set, we write the set by some Sets are denoted by capital letters A, B, C etc., while special property or properties satisfied by all elements and the elements are denoted in general by small letters a, b, c, writeitas, A= { x : P ( x )} etc. where P() x isapropertywhichissatisfiedby x. Followingcollectionsaresets e.g., A = {,,,}1 2 3 4 ={:x x ∈ N and x < 5} (i) Thecollectionofallpositiveintegers. This method is also known as Rule method or Property (ii) ThecollectionofallcapitalsofstatesofIndia. method. Let A be any set of objects and let ‘a’ be a member of A, % In writing the elements of any set there is no consideration of then we write a∈ A and read it as ‘a belong to A’ or ‘a is an sequence, e.g.,{,,,,}a e i o u and {,,,,}e i o u a aretwosamesets. element of A’ or ‘a is member of A’. If a is not an object of A, then we write a∉ Aand read as ‘a does not belong to A’ or ‘a isnotanelement of A’or‘a isnotmemberof A’. TypesofSets Somestandardnotationsforsomespecialpoints. (i) Empty set A set consisting of no element is (i) The set of all natural numbers i.e., the set of all called an empty set or null set or void set and is denoted by φ positiveintegersisdenotedby N. symbol or { }. = ∈ < < } = φ (ii) Thesetofallintegersisdenotedby Z or I. e.g., A{ x : x N and 3 x 4 . (iii) Thesetofallrationalnumbersisdenotedby Q. Asetwhichisnotemptyiscallednon-emptysetor non-voidset. (iv) Thesetofallrealnumbersisdenotedby R. + (ii) Singleton set A set consisting of only one (v) The set of all positive real numbers is denoted by R . elementiscalledasingletonset. (vi) Thesetofallcomplexnumbersisdenotedby C. e.g., {2},{0},{φ} (vii) The set of all positive rational numbers is denoted by Q+. % Theset{0}isnotanemptysetasitcontainsoneelement0. % Theset{φ}isnotanemptysetasitcontainsoneelement φ. RepresentationofSets (iii) Finite set A set having finite number of Therearetwomethodstorepresentsets elementsiscalledfiniteset. e.g., A = {1,2,3}isafiniteset. (i)Listingmethod(ii)Setbuildermethod 2 NDA/NA Mathematics (iv) Infinite set A set which is not finite is called % Whenever, we have to show that two sets A and B are equal i.e., = ⇔ ⊆ ⊆ aninfiniteset. A B ABBAand . = e.g., A Setofpointslieinaplaneisaninfiniteset. (x) Universal set In any discussion in set theory, (v) Cardinal number of a finite set The we need a set such that all sets under consideration in that number of elements of a finite set A is called its cardinal discussion are its subsets. Such a set is called the universal numberanditisdenotedby n(A)or o(A). set and is denoted by U. (vi) Equivalent sets Two finite sets A and B are (xi) Power set The set of all the subsets of a given said to be equivalent if they have the same cardinal set A is said to be the power set of A and is denoted by P(A ). number.Thus,sets A and B areequivalentif n()()AB= n . e.g., If A = {,,}1 2 3 ,then (vii) Subset and super set The set B is said to PA( )= { φ ,{1 },{ 2 },{ 3 },{ 12 , },{ 23{,},{,,}} , }, 3 1 1 2 3 . be subset of set A, if every element of set B is also an % Elementsofpowersetarethesubsetof A. ⊆ element of set A. Symbolically we write it as, BA or % Thepowersetofeachgivensetisalwaysnon-empty. AB⊇ ,where A issupersetof B. % If A is a finite set of n elements, then number of elements in PA() n (a) BA⊆ is read as B is contained in A or B is subset willbe 2 . of A or A issupersetof B. (b) AB⊇ is read as A contains B or B is a subset of A. VennDiagram Evidently, if A and B are two sets such that To express the relationship among sets in a ∈ ⇒ ∈ ⇒ x B x A, then B is subset of A. The symbol ‘ ’ stands perspective way, we represent them pictorially by means of for ‘implies’, we read it as x belongs to B implies that x diagrams,knownasVenndiagrams. belongsto A. e.g.,Let A = {1 , 2 , 3 , 4 }; B = {,,}1 2 4 U Here, B isasubsetof A. (viii) Proper subset The set B is said to be a proper subset of set A, if every element of set B is an element of A whereas every element of A is not an element The universal set is usually represented by a of B. rectangular region and its subsets by circle or closed We write it as BA⊂ and read it as ‘B is a proper subset boundedregionsinsidethisrectangularregion. of A’. Thus, B is a proper subset of A, if every element of B is an element of A and there is atleast one element in A which OperationsonSets is not in B. (i) Union of sets Let A and B are two sets, then Observe that A⊆ A i.. e, every set is a subset of itself, union of A and B is denoted by AB∪ and it consists of each butnotapropersubset. oneofwhichiseitherin A orin B orinboth A and B. e.g., Let AB={1 , 2 , 3 }; = { 1 , 2 }, then BA⊂ . U % Nullsetisasubsetofeverysetandeachsetissubsetofitself. % Numberofsubsetofafinitesetof n elementsis 2n. A B (ix) Equal sets Two sets A and B are said to be equal, if each element of A is an element of B and each Thus, A∪ B ={ x:} x ∈ Aor x ∈ B elementof B isanelementof A. Clearly, x∈ A ∪ B ⇔ x ∈ Aor x ∈ B Thus, two sets A and B are equal, if they have exactly and x∉ A ∪ B ⇔ x ∉ Aand x ∉ B the same elements but the order in which the elements in Inthefigure,theshadedpartrepresents AB∪ . thetwosetshavebeenwrittendownisimmaterial. Itisevidentthat AABBAB⊆ ∪,. ⊆ ∪ Thus,if x∈ A ⇒ x ∈ B and y∈ B ⇒ y ∈ A, Example 1. If A = {,,}1 2 3 and B = {,,,}1 3 5 7 , then the ∪ then A and B areequal valueof AB is (a){4,5,7} (b){1,2,3,5,7} e.g., {4,8,10}={8,4,10} (c){1,2,3,5} (d){6,3,5,7} [The order in which the elements of a set is also Solution (b) Q AB={}{}1, 2 , 3 and = 1 , 3 , 5 , 7 immaterial] ∴ AB∪ = {}1,,,, 2 3 5 7 Sets,RelationsandFunctions 3 (ii) Intersection of sets The intersection of two (v) Symmetric difference of two sets The sets A and B, denoted by AB∩ is the set of all elements, symmetric difference of two sets A and B, denoted by AB∆ common to both A and B. is the set (ABBA− ) ∪ ( − ). U U A–B B–A AB A B ∩ = ∈ ∈ Thus, A B{ x: x A and x B} Thus, ABABBA∆ =()() − ∪ − ={:x x ∉ A ∩ B} ∈ ∩ ⇔ ∈ ∈ Clearly, x A B x A and x B Theshadedpartrepresents AB∆ . and x∉ A ∩ B ⇔ x ∉ A or x∉ B = = Inthefigure,theshadedpartrepresents AB∩ . Example 3. If A {,,,,}1 3 5 7 9 and B {,,,,}2 3 5 7 11 , Itisevidentthat ABAABB∩ ⊆,. ∩ ⊆ thenfindthevalueof AB∆ is (a){5,7,11} (b){1,2,9,11} Example 2. If A = {,,,}1 2 3 4 and B = {,,}2 4 6 , then the (c){3,5,7} (d){1,3,5,11} ∩ valueof AB is Solution (b) Q A = {}1,,,, 3 5 7 9 and B ={,,,,}2 3 5 7 11 (a){2,4} (b){6,5} (c){2,3,4} (d){1,2,3} ∴ AB− ={ , }19 and BA− ={,}2 11 ∴ ABABBA∆ =()() − ∪ − Solution (a)Q A = {,,,}1 2 3 4 and B ={,,}2 4 6 ={,}{,}1 9 ∪ 2 11 ={,,,}1 2 9 11 ∴ AB∩ = {,}2 4 (iii) Disjoint sets Two sets U ComplementofaSet A and B are said to be disjoint sets, if they have no common element A B If U is a universal set and AU⊂ ,then complement set ′ − i.e., AB∩ = φ. of A isdenotedby AUAor . The disjoint sets can be U representedbyVenndiagramasshowninthefigure A B e.g.,let A = {,,}1 2 3 and B = {,}4 6 Here, A and B aredisjointsetsbecause AB∩ = φ . (iv) Difference of sets If A and B are two sets, Thus, A′ = U − A ={ x:,} x ∈ U but x ∉ A then their difference AB− is the set of all those elements Itisclearthat x∈ A ′ ⇔ x ∉ A of A whichdonotbelongto B. % φ =U ′ % φ′ = U % ()AA′ ′ = U % AAU∪ ′ = % AA∩ ′ = φ A–B LawsofAlgebraofSets AB If A,B and C arethreesets,then Thus, A− B ={:} x x ∈ Aand x ∉ B 1. Idempotent laws ∈ − ⇔ ∈ ∉ Clearly, x A B x Aand x B. (a) AAA∪ = (b) AAA∩ = − Inthefigure,theshadedpartrepresents AB. 2. Identitylaws Similarly, the difference BA− is the set of all those (a) AA∪ φ = (b) AUA∩ = elementsof B thatdonotbelongto A i.e., 3. Commutativelaws B− A ={:} x x ∈ Band x ∉ A (a) ABBA∪ = ∪ (b) ABBA∩ = ∩ U 4. Associativelaws B–A (a) ()()ABCABC∪ ∪ = ∪ ∪ (b) ABCABC∩()() ∩ = ∩ ∩ B A 5. Distributivelaws Inthefigure,theshadedpartrepresents BA− . (a) ABCABAC∪()()() ∩ = ∪ ∩ ∪ ∩ ∪ = ∩ ∪ ∩ e.g., if A = {,,,,}1 3 5 7 9 and B = {2 , 3 , 5 , 7 , 11 }, then (b) ABCABAC()()() A −B = {,}1 9 and BA− = {2 , 11 }. 6. De-Morgan’slaws (a) ()ABAB∪ ′ = ′ ∩ ′ (b) ()ABAB∩ ′ = ′ ∪ ′ 4 NDA/NA Mathematics 7. (a) ABAB− = ∩ ′ (b) BABA− = ∩ ′ Facts Related to Cartesian Product (c) ABAAB− = ⇔ ∩ = φ (d) ()ABBAB− ∪ = ∪ (e) ()ABB− ∩ = φ If A, B and C are non-empty sets, then (f) ()()()()ABBAABAB− ∪ − = ∪ − ∩ 1.