Sets

Sets & Relations SETS SET A set is a collection of well defined objects which are distinct from each other. Set are generally denoted by capital letters A, B, C, ...... etc. and the elements of the set by small letters a, b, c ..... etc. If a is an element of a set A, then we write a  A and say a belongs to A. If a does not belong to A then we write a  A, e.g. the collection of first five prime natural numbers is a set containing the elements 2, 3, 5, 7, 11. METHODS TO WRITE A SET : (i) Roster Method or Tabular Method : In this method a set is described by listing elements, separated by commas and enclose then by curly brackets. Note that while writing the set in roster form, an element is not generally repeated e.g. the set of letters of word SCHOOL may be written as {S, C, H, O, L}. (ii) Set builder form (Property Method) : In this we write down a property or rule which gives us all the element of the set. A = {x : P(x)} or {x / P(x)} where P(x) is the property by which x  A and colon ( : ) or Slash ( /) stands for ‘such that’ Ex.1 Express set A = {x : x  N and x = 2n for n  N} in roster form Sol. A = {2, 4, 6,.........} Ex.2 Express set B = {x2 : x < 4, x  W} in roster form Sol. B = {0, 1, 4, 9} Ex.3 Express set A = {2, 5, 10, 17, 26} in set builder form Sol. A = {x : x = n2 + 1, nN, 1  n  5} TYPES OF SETS Null set or empty set : A set having no element in it is called an empty set or a null set or void set, it is denoted by  or { }. A set consisting of at least one element is called a non-empty set or a non-void set. Singleton set : A set consisting of a single element is called a singleton set. Finite set : A set which has only finite number of elements is called a finite set. Order of a finite set : The number of elements in a finite set A is called the order of this set and denoted by O(A) or n(A). It is also called cardinal number of the set. e.g. A = {a, b, c, d}  n(A)= 4 1 Sets & Relations Infinite set : A set which has an infinite number of Note : elements is called an infinite set. (i) Every set is a subset of itself Equal sets : Two sets A and B are said to be equal (ii) Empty set  is a subset of every set if every element of A is member of B, and every   element of B is a member of A. If sets A and B are (iii) A B and B A  A = B equal, we write A = B and if A and B are not equal (iv) The total number of subsets of a finite set containing n elements is 2n. then A  B Equivalent sets : Two finite sets A and B are (v) Number of proper subsets of a set having n elements equivalent if their number of elements are same is 2n – 1. i.e. n(A) = n(B) (vi) Empty set  is proper subset of every set except e.g. A = {1, 3, 5, 7}, B = {a, b, c, d}  n(A) = 4 and n(B) = 4  A and B are equivalent sets Note - Equal sets are always equivalent but equivalent sets may not be equal itself. POWER SET : Let A be any set. The set of all subsets of A is called power set of A and is denoted by P(A) X = {x 1 , x 2 , x 3 , ..........x n} then n(P(X)) = 2 n ; n(P(P(x))) = 2 2n Ex.4 Identify the type of set : (i) A = {x  N : 5 < x < 6} Ex.5 Examine whether the following statements are true or false : (ii) A = {a, b, c} (iii) A = {1, 2, 3, 4, .......} (i) {a, b} (iv) A = {1, 2, 6, 7} and B = {6, 1, 2, 7, 7} (v) A = {0} Sol. (i) Null set (iii) infinite set (ii) finite set (iv) equal sets (v) singleton set / {b, c, a} (ii) {a, e} / {x : x is a vowel in the English alphabet} (iii) {1, 2, 3}  {1, 3, 5} (iv) {a} {a, b, c} Sol. (i) False as {a, b} is subset of {b, c, a} (ii) True as a, e are vowels SUBSET AND SUPERSET : (iii) False as element 2 is not in the set {1, 3, 5} (iv) False as a {a, b, c} and {a}  {a, b, c} Let A and B be two sets. If every element of A is an element B then A is called a subset of B and B is Ex.6 If a set A = {a, b, c} then find the number of subsets of the set A and also mention the set of called superset of A. We write it as A  B. e.g. A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6, 7} all the subsets of A.  A  B Sol. Since n(A) = 3 If A is not a subset of B then we write A / B  number of subsets of A is 23 = 8 PROPER SUBSET : If A is a subset of B but A  B then A is a proper subset of B and we write A  B. Set A is not proper subset of A so this is improper subset of A and set of all those subsets is P(A) named as power set P(A): {, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}} 2 Ex.7 Find power set of set A = {1, 2} Sol. P(A) = {, {1}, {2}, {1, 2}} Ex.8 If  denotes null set then find P(P(P())) Sol. Let P() = {} P(P()) = {,{}} P(P(P())) = {, {}, {{}}, {, {}}} Sets & Relations SOME OPERATION ON SETS : (i) Union of two sets : If A and B are two sets then union () of A and B is the set of all those elements which belong either to A or to B or to both A and B. A  B = {x : x  A or x  B} e.g. A = {1, 2, 3}, B = {2, 3, 4} then A  B = {1, 2, 3, 4} UNIVERSAL SET : A set consisting of all possible elements which occur AB in the discussion is called a universal set i.e. it is a (ii) Intersection of two sets : If A and B are two sets super set of each of the given set. Thus, a set that then intersection () of A and B is the set of all those contains all sets in a given context is called the elements which belong to both A and B. universal set. It is denoted by U. e.g. if A = {1, 2, 3}, A  B = {x : x  A and x  B} B = {2, 4, 5, 6}, C = {1, 3, 5, 7} then U = {1, 2, 3, 4, 5, 6, 7} can be taken as the universal set. e.g. A = {1, 2, 3}, B = {2, 3, 4} then A  B = {2, 3} VENN (EULER) DIAGRAMS The diagrams drawn to represent sets are called Venn diagram or Euler-Venn diagrams. Here we represents A B the universal U as set of all points within rectangle (iii) Difference of two sets : If A and B are two sets and the subset A of the set U is represented by the then the difference of A and B, is the set of all those interior of a circle. If a set A is a subset of a set B, elements of A which do not belong to B. then the circle representing A is drawn inside the circle ` A – B = {x : x  A and x  B}. It is also written as representing B. If A and B are not equal but they AB'. have some common elements, then to represent A Similarly B – A = B  A' e.g. A = {1, 2, 3}, and B by two intersecting circles. B = {2, 3, 4} ; A – B = {1} e.g. If A is subset of B then it is represented diagrammatically in fig. U B A e.g. If A is a set then the complement of A is represented in fig. A– B B–A (iv) Symmetric difference of sets : Set of those elements which are obtained by taking the union of the difference of A & Bis (A – B) & the difference of B &A is (B – A), is known as the symmetric difference of two sets A & B. It is denoted by A  B and A  B = (A – B)  (B – A) U A A (A B) = (A – B)  (B – A) 3 Sets & Relations (v) Complement of a set : Complement of a set A is a set containing all those elements of universal set which are not in A. It is denoted by A ,AC or A. So AC = {x : x  U but x  A}. e.g. If set A = {1, 2, 3, 4, 5} and universal set U = {1, 2, 3, 4,.......50} then A = {6, 7, .......50} (iii) Commutative Law : For any set A and B, we have (i) A  B = B  A and (ii) A  B = B  A i.e. union and intersection are commutative. (iv) Associative Law : If A, B and C are any three sets then (i) (A  B)  C = A  (B  C) A’ NOTE : All disjoint sets are not complementary sets but all complementary sets are disjoint. (ii) (A  B)  C = A  (B  C) i.e. union and intersection are associative. (v) Distributive Law : If A, B and C are any three sets then (vi) Disjoint sets : If A  B = , then A, B are disjoint (i) A  (B  C) = (A  B)  (A  C) e.g. If A = {1, 2, 3}, B = {7, 8, 9} then A  B =  (ii) A  (B  C) = (A  B)  (A  C) i.e. union and intersection are distributive over intersection and union respectively. Disjoint LAWS OF ALGEBRA OF SETS (PROPERTIES OF SETS): (i) Idempotent Law : (vi) De-Morgan’s Principle : If A and B are any two sets, then (i) (A  B) = A  B (ii) (A  B) = A  B For any set A, we have (i) A  A = A and (ii) A  A = A Proof : (i) Let x be an arbitrary element of (A  B) . Then x  (A  B)  x  (A  B) Proof :  x  A and x  B (i) A  A = {x : x  A or x  A} = {x : x  A}=A Again let y be an arbitrary element of A  B. Then y  A  B (ii) A  A = {x : x  A & x  A} = {x : x  A}=A  x  A  B  y  A and y  B  y  A and y  B For any set A, we have  y  (A  B)  y  (A  B) (i) A   = A and  A  (ii) A  U = A i.e.  and U are identity elements for union and intersection respectively Hence (A  B) = A  B (ii) Identity Law : B  (A  B). Similarly (ii) can be proved. Proof : (i) A   = {x : x  A or x  } = {x : x  A} = A (ii) A  U = {x : x  A and x  U} = {x : x  A} = A NOTE : (i) A – (B  C) = (A – B)  (A – C) ; A – (B  C) = (A – B)  (A – C) (ii) A   = , A  U = U 4 Sets & Relations Ex.16 If A = {2, 3, 4, 5, 6, 7} and B = {3, 5, 7, 9, 11, Ex.9 Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12} then find A  B 13} then find A – B and B – A.. Sol. A – B = {2, 4, 6} & B – A = {9, 11, 13) Sol. A  B = {2, 4, 6, 8, 10, 12} Ex.10 Let A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}. Find A – B and B – A. Sol. A – B = {x : x  A and x  B} = {1, 3, 5} similarly B – A = {8} Ex.11 State true or false : (i) A  A =  SOME IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS : If A, B & C are finite sets and U be the finite universal set, then (ii)  A = A (i) n(AB) = n(A) + n(B) – n(A B) Sol. (i) false because A  A' = U (ii) true as  A = U A = A (ii) n(AB) = n(A) + n(B) (if A & B are disjoint sets) Ex.12 Use Venn diagram to prove that A  B  B  A. (iv) n(A B) = n[(A – B)  (B – A)] (iii) n(A – B) = n(A) – n(A B) = n(A) + n(B) – 2n(A  B) (v) n(A  B  C) = n(A) + n(B) + n(C) – n(A B) Sol. – n(B  C) – n(A  C) + n(A  B  C) From venn diagram we can conclude that B  A. Ex.13 Prove that if A  B = C and A  B =  then A = C – B. Sol. Let x A xA B xC (vii) n(A B) = n(A  B) = n(U) – n(A  B) ( AB = C) Now A B =   x  B ( x  A)  x C – B ( x  C and x  B)  A  C – B Let x  C – B (vi) n(A B) = n(A  B) = n(U) – n(A  B)  x  C and x  B x  A  B and x  B  x  A  C – B A A=C – B  Ex.14 If A = {x : x = 2n + 1, n  Z and B = {x : x = 2n, n  Z}, then find A  B. Sol. A B = {x : x is an odd integer}  {x : x is an even integer} = {x : x is an integer} = Z Ex.15 If A = {x : x = 3n, n  Z} and B = {x : x = 4n, n  Z} then find A  B. Sol. We have, x  A  B  x = 3n, n  Z and x = 4n, n  Z  x is a multiple of 3 and x is a multiple of 4  x is a multiple of 3 and 4 both  x is a multiple of 12  x = 12n, n  Z Hence A  B = {x : x = 12n, n  Z} Ex.17 In a group of 40 students, 26 take tea, 18 take coffee and 8 take neither of the two. How many take both tea and coffee ? Sol. n(U) = 40, n(T) = 26, n(C) = 18 n(T  C) = 8  n(T  C) = 8  n(U) – n(T  C) = 8  n(T  C) = 32  n(T) + n(C) – n(T  C) = 32  n(T  C) = 12 Ex.18 In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find (i) How many drink tea and coffee both ? (ii) How many drink coffee but not tea ? 5 Sets & Relations Sol. T : people drinking tea Case-I C : people drinking coffee When n(A  B) is minimum, i.e., n(A  B) = 0 (i) n(T) = n(T – C) + n(T  C)  30 = 14 + n(T  C) n(T  C) = 16 This is possible only when A  B = . In this case, n(A  B) = n(A) + n(B) – 0 = n(A) + n(B) = 3 + 6 = 9. So, maximum number of elements in A  B is 9. Case-II (ii) n(C – T) = n(T  C) – n(T) = 50 – 30 = 20 Ex.19 If A and B be two sets containing 3 and 6 elements respectively, what can be the minimum number of elements in A  B? Find also, the maximum number of elements in A  B. Sol. We have, n(A  B) = n(A) + n(B) – n(A  B). This shows that n(A  B) is minimum or maximum according as n(A  B) is maximum or minimum respectively. When n(A  B) is maximum. This is possible only when A  B. In this case, n(A  B) = 3  n(A  B) = n(A) + n(B) – n(A  B) = (3 + 6 – 3) = 6 So, minimum number of elements in A  B is 6. 6