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Set theory

- 2. SET A ‘Set’ is any well defined collection of objects called the elements or members of the set. Following are some examples of sets: • The collection of all planets in our universe. • The collection of all states in India. • The collection of vowels in English alphabets, i.e. A = {a, e, i, o, u}. • The collection of first five natural numbers. A = {1,2,3,4,5} x S means “x is an element of set S.” x S means “x is not an element of set S.”
- 3. {1, 2, 3} is the set containing “1” and “2” and “3.” {1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant. {6, 11} = {11, 6} = {11, 6, 6, 11} {1, 2, 3} = {3, 2, 1} since sets are unordered. {1, 2, 3, ..., 100},where the ellipsis ("...") indicates that the list continues in the obvious way. Ellipsis may also be used where sets have infinitely many members. Thus, the set of positive even numbers can be written as {2, 4, 6, 8, ... }. {0,1, 2, 3, …} is a way we denote an infinite set (in this case, the natural numbers). NOTE:
- 4. SOME IMPORTANT SETS: • N : For the set of natural numbers • Z or I : For the set of integers • Z+ or I+ : For the set of all positive integers • Q : For the set of all rational numbers • Q+ : For the set of all positive rational numbers • R : For the set of all real numbers • R+ : For the set of all positive real numbers • C : For the set of all complex numbers
- 5. REPRESENTATION OF A SET A set is often represented in the following two ways: (I) Roster form (Tabular form) -set is described by listing elements separated by commas, within braces { }. Eg: the set of even natural numbers {2, 4, 6, 8, ...}. (II)Set Builder form( Rule form) - set is described by a characterizing property P(x) of its element x. {x : P(x) holds} or {x / P(x) holds} The symbol ‘|’ or ‘:’ is read as ‘such that’. Eg: 1) Set of vowels of English Roster form :- A= {a,e,i,o,u} Set Builder Form:- A= {x: x is an English vowel} 2) Set of first ten natural numbers Roster form :- A= {1,2,3,4,5,6,7,8,9,10} Set Builder Form:- A= {x: x N and 1≤x ≤10} 3) {x: x is an integer and 4 < x ≤8}={5,6,7,8} 4) {x: x is a month with exactly 30 days}={April,June,September,November}
- 6. TYPES OF SETS Empty set: A set containing no element is called an empty set. It is also known as null set or void set. It is denoted by {},Φ Eg: B set of immortal man (b) 2 A x R / x 10 (a) Singleton set: A set having exactly one element is called singleton set. For example, {2}, {0}, {5} are singleton sets. A={x:x is a solution of the equation 2x-5=0} 2 5
- 7. Finite set: A set consisting of a finite number of elements is called a finite set. Eg: •{1, 2, 4, 6} is a finite set because it has four elements. •The set of months is a finite set because it has 12 elements. Infinite set: A set which is not a finite set, i.e., a set consisting of infinite number of elements is called an infinite set. Eg: (i) The set of all straight line in a given plane. (ii) The set of all natural numbers. (iii) The set of real numbers between ‘1’ and ‘2’.
- 8. CARDINAL NUMBER OF A SET The number of elements in a finite set is called cardinal number or order of the set. It is referred as the cardinality of the set A and is denoted by n(A),ǀAǀ or #(A). For example, if A 1 , 2, 3, 4, 5 n A 5 or o A 5. Eg: S = { {1, 2}, {2, 4}, {3, 5, 9} } Set of sets: A set S having all its elements as sets is called set of sets. 3 S But S = { {1, 2}, 4, {3, 5, 9} } is not a set of sets as element 4 ϵ S is not a set. If S = {3,3,3,3,3}, |S| = 1 If S = then |S| = 0.
- 9. EQUIVALENT AND EQUAL SETS Equivalent sets: Two finite sets A and B are equivalent if their cardinal number is same, i.e. n(A) = n(B). Equal sets: Two sets A and B are said to be equal if every element of A is a member of B, and every element of B is a member of A. For example: A = {4, 5, 6} and B = {a, b, c} are equivalent ie. A Ξ B but A = {4, 5, 6} and C = {6, 5, 4} are equal, ie. A = C.
- 10. Subsets: A set A is said to be a subset of a set B, written as A B, if each element of A is also an element of B. A B, if x A x B A Subsets Eg: Let A = {2, 4, 6, 8}, B = {2, 4, 6, 8, 10, 12}, then B A: B is a superset of A B contains A A B.
- 11. Proper subset: A set A is said to be a proper subset of a set B if every element of A is an element of B and B has at least one element which is not an element of A. A B means “A is a proper subset of B.” A B, and A B. A subset which is not proper is called improper subset. For example: Let A = {1, 2, 3}, B = {2, 3, 4, 1, 5}, then A B Thus, if A is a proper subset of B, then there exists an element such that x B x A. For example, {1} 1 , 2, 3 but 1 , 4 1 , 2, 3 . {1,2,3} {1,2,3,4,5} {1,2,3} {1,2,3,4,5}
- 12. SOME RESULTS ON SUBSETS (i) Every set is a subset of itself. (ii) The empty set is a subset of every set. (iii) The total number of subsets of a finite set containing n elements is 2n. Proof : We know that nCr denotes the number of ways for choosing r things from n different things. Therefore each selection of r things gives a subset of the set A containing r elements. The number of subsets of A having no element nC0 The number of subsets of A having one element nC1
- 13. The number of subsets of A having two elements = nC2 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The number of subsets of A having n elements = nCn. Hence, the total number of subsets of A n n n n n 0 1 2 n C C C ... C 2 [ We know that Putting x = 1, we get ] n n n n 2 n n 0 1 2 n 1 x C C x C x ... C x . n n n n n 0 1 2 n 2 C C C ... C .
- 14. POWER SET The set of all the subsets of a given set A is said to be the power set of A and is denoted by P(A). i.e. P A S|S A S P A S A Also, for all sets A. P A and A P A The power set of a finite set with n elements has 2n elements. For example, if A = {a, b, c}, then P(A) , a , b , c , a, b , a, c , b, c , a, b, c . P(A) A
- 15. UNIVERSAL SET Any set which is super set of all the sets under consideration is called the universal set and is denoted by or . For example: (i) When we are using sets containing natural numbers then N is the universal set. Euler-Venn Diagram- a pictorial representation of sets A B A B
- 16. SET OPERATIONS A B x:x A or x B . Union of sets: The union of A and B, denoted by A ∪ B, is the set of all things which are members of either A or B. Clearly, if and x A B x A or x B x A B x A and x B A B A B (a) Generalised definition If is a finite family of sets, then their union is 1 2 n A , A , ...A n 1 2 3 n n i 1 A A A ...UA or A . •{1, 2} ∪ {red, white} ={1, 2, red, white}. •If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A B = {Charlie, Lucy, Linus, Desi}
- 17. Intersection of sets: The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. A B x : x A and x B Clearly, if and x A B x A and x B x A B x A or x B. B A B A A B A B Generalised definition The intersection of sets is the set of all the elements which are common to all the sets It is denoted by 1 2 n A , A , ...A n 1 2 3 n i i 1 A A A ... A or A 1 2 n A , A , ... A . •If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A B = {Lucy} •{1, 2} ∩ {red, white} = ∅. •{1, 2, green} ∩ {red, white, green} = {green}. •{1, 2} ∩ {1, 2} = {1, 2}.
- 18. A B x:x A and x B DIFFERENCE OF SETS x B A x : x B and x A A B When neither A B nor B A (c) A B When neither A B nor B A (c) The difference of two sets A and B, denoted by A-B or A B, is a set which contains those elements of A which are not in B. Eg: If A={1,2,3,4} and B={3,4,5,6} then A-B={1,2} B-A={5,6} A B A-B B-A
- 19. Disjoint sets: Sets whose intersection is empty are called disjoint sets A B A B A B Complement of set (A’, Ac, A ) A A ´ c A or A x : x U and x A x A x A A’ A’=U-A= Eg: If U={1,2,3,4,5} and A={12,3} Then, A’=U-A = {4,5} Eg: Let A={a,b,c,d,e} and B= {x,y,z} Then, clearly A B
- 20. The symmetric difference of sets A and B , denoted by A B or A Δ B, consists of those elements which belong to either A or B but not both. ie. A B = { x : (x A and x B) or (x B and x A)} SYMMETRIC DIFFERENCE OF SETS = (A – B) (B – A) A B A B B A (b) A B When neither nor B B A A x A B x A B (A U B)-(A B) Eg: If A={1,2,3,4} and B={3,4,5,6} then A-B={1,2} B-A={5,6} A B = {1,2,5,6} = (A – B) (B – A) A B
- 21. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B. A B = { (a, b) : a A and b B} Let A and B be finite sets, then CARTESIAN PRODUCT A1 A2 … An = {(a1, a2,…, an): a1 A1, a2 A2, …, an An} Eg: •{1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2, white)}. •{1, 2} × {a, b} = {(1, a), (1, b), (2, a), (2, b)}. | A × B | = | B × A | = | A | × | B |.
- 22. A unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a function f :A→ A where A is a set . In this case f is called a unary operation on A. Common notations are prefix (e.g. +, −, not), postfix (e.g. n!), functional notation (e.g. sin x or sin (x)), and superscripts (e.g. complement A, transpose AT). A binary operation is a calculation involving two operands. Let A, B and C be three sets . Then a relation * from A × B → C is a binary relation. a*b=c Typical examples of binary operations are the addition (+) and multiplication (×) of numbers and matrices as well as composition of functions on a single set. In sets: unary operation: complement binary operation : union, intersection, difference, symmetric BINARY AND UNARY OPERATION
- 23. REMARK: (i) U x:x U and x U . (ii) x:x U and x U. (iii) A x:x U and x A x : x U and x A A (iv) A A x:x U and x A x:x U and x A U (v) A A x:x U and x A x:x U and x A
- 24. PROCEDURE FOR PROVING EQUALITY OF SETS As we have discussed earlier that two sets A and B are said to be equal if every element of set A is an element of set B and every element of B is an element of A. It is clear that A B A B and B A. i.e. A B x A x B .
- 25. ALGEBRA OF SETS Idempotent laws For any set A, we have (i) (ii) A A A A A A Identity laws For any set A, we have (i) (ii) A A A U A Involution Laws (A’)’= A Complement Laws (i) A U A’=U (ii) A A’ = Φ (iii) U’ =Φ (iv) Φ’ = U
- 26. Commutative laws For any two sets A and B, we have (i) (ii) A B B A A B B A i.e. union and intersection are commutative. Proof: As we know that two sets X and Y are equal if X Y and Y X. (i) Let x be any arbitrary element of . A B x A B x A or x B x B or x A x B A A B B A ...(i) (ii) Similarly, let y be any arbitrary element of B A y B A y B or y A y A or y B y A B B A A B ...(i) …(ii) From (i) and (ii) A B B A
- 27. Associative laws If A, B and C are any three sets, then (i) (ii) i.e. union and intersection are associative. A B C A B C A B C A B C Proof: (i) Let x be any arbitrary element of . A B C x A B C x A B or x C x A or x B or x C x A or x B or x C x A or x B C x A B C A B C A B C ...(i)
- 28. (ii) Similarly, let y be any arbitrary element of . A B C y A B C y A or y B C y A or y B or y C y A or y B or y C y A B or y C y A B C A B C A B C ...(ii) From (i) and (ii) A B C A B C Proved.
- 29. Distributive laws ie, union and intersection are distributive over intersection and union respectively. If A, B and C are any three sets, then (i) (ii) A B C A B A C A B C A B A C Proof: (i) Let x be any arbitrary element of . A B C x A B C x A or x B C x A or x B and x C x A or x B and x A or x C [ ‘or’ is distributive over ‘and’] x A B and x A C x A B A C A B C A B A C ...(i)
- 30. (ii) Similarly, let y be any arbitrary element of . A B A C y A B A C y A B and y A C y A or y B and y A or y C y A or y B and y C y A or y B C y A B C A B A C A B C ...(ii) From (i) and (ii), A B C A B A C Proved.
- 32. DE MORGAN’S LAW If A and B are any two sets, then (i) (ii) A B A B A B A B Proof: (i) Let x be an arbitrary element of the set . A B x A B x A B x A and x B x A and x B x A B A B A B ...(i) (ii)Now, let y be an arbitrary element of A B y A B y A and y B y A and y B y A B y A B A B A B ' ...(ii) From (i) and (ii), A B A B
- 33. First, let A = B. Then A B A and A B A A B A B A B A B A B ...(i) Conversely, let A B A B. x A x A B x A B x A and x B x B Q: For any two sets A and B, prove that A B A B A B. A B ...(ii) Now let y B y A B y A B y A and y B y A B A ...(iii) From (ii) and (iii), we get A = B Thus, A B A B A B ...(iv) From (i) and (iv), A B A B A B
- 34. Sol:We have aN ax:x N 3N 3x:x N 3, 6, 9,12,15, ... 7N 7x:x N 7,14, 21 , 28, ... Hence, 3N 7N 21 , 42, 63, ... 21x:x N 21N Note that where c = LCM of a, b. aN bN cN Q: If such that , describe the set a N aN ax:x N 3N 7N.
- 35. Let x be any element of . A B C x A B C x A and x B C x A and x B or x C x A and x B or x A and x C x A B or x A C x A B A C A B C A B A C ...(i) Q:If A, B and C are any three sets, then prove that A B C A B A C . Again y be any element of . A B A C y A B A C y A B or y A C y A and y B or y A and y C y A and y B or y C y A and y B C y A B C A B A C A B C ...(ii) From (i) and (ii), A B C A B A C Proved.
- 36. Wehave A B C. C B A B B A B B X Y X Y A B B B [By distributive law] A B A B = A – B = A A B Proved. Q: Let A, B and C be three sets such that and . then prove that A = C – B A B C A B .
- 37. Let x be any arbitrary element of C – B. x C B x C and x B x C and x A A B x C A C B C A Proved. If A, B and C are the sets such that then prove that A B, C B C A.
- 38. PROBLEMS Q.1. A survey was conducted, it was found that the number of people who prefer only Burger, only Pizza, both Burger and Pizza and neither of them are 40, 45, 18 & 22 respectively. Find the number of people surveyed. A. 69 B. 79 C. 82 D. 89
- 39. Q.2.There are 300 students in a class 10th. Among them, 80 students are learning both History & Geography. A total of 180 students are learning History. If every student is learning at least one of these two subjects, how many students are learning Geography in total.
- 40. Q.3. In class 12th, 70% of the students study Accountancy and 40% of the students study Business Studies. If 15% of the total students study both Accountancy and Business Studies, what % of the students do not study either of the two subjects?
- 41. Q.4. Among a group of students, 50 played cricket, 50 played hockey and 40 played volley ball. 15 played both cricket and hockey, 20 played both hockey and volley ball, 15 played cricket and volley ball and 10 played all three. If every student played at least one game, find the number of students and how many played only cricket, only hockey and only volley ball?