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SETS, RELATIONS & FUNCTIONS, , SETS, RELATIONS & FUNCTIONS, SETS, 2.2 Set-Builder Form, , 1. SET, A set is a collection of well-defined and well distinguished, objects of our perception or thought., 1.1 Notations, The sets are usually denoted by capital letters A, B, C, etc., and the members or elements of the set are denoted by lowercase letters a, b, c, etc. If x is a member of the set A, we write, x A (read as 'x belongs to A') and if x is not a member of the, set A, we write x A (read as 'x does not belong to A,). If x, and y both belong to A, we write x, y A., , 2. REPRESENTATION OF A SET, Usually, sets are represented in the following two ways :, (i), , Roster form or Tabular form, , (ii), , Set Builder form or Rule Method, , 2.1 Roster Form, In this form, we list all the member of the set within braces, (curly brackets) and separate these by commas. For example,, the set A of all odd natural numbers less that 10 in the Roster, form is written as :, A = {1, 3, 5, 7, 9}, , In this form, we write a variable (say x) representing any, member of the set followed by a property satisfied by each, member of the set., For example, the set A of all prime numbers less than 10 in, the set-builder form is written as, A = {x | x is a prime number less that 10}, The symbol '|' stands for the words 'such that'. Sometimes,, we use the symbol ':' in place of the symbol '|'., , 3. TYPES OF SETS, 3.1 Empty Set or Null Set, A set which has no element is called the null set or empty, set. It is denoted by the symbol I ., , For example, each of the following is a null set :, (a), , The set of all real numbers whose square is –1., , (b), , The set of all rational numbers whose square is 2., , (c), , The set of all those integers that are both even and odd., A set consisting of atleast one element is called a, non-empty set., , 3.2 Singleton Set, A set having only one element is called singleton set., For example, {0} is a singleton set, whose only member is 0., (i), (ii), , In roster form, every element of the set is listed, only once., , 3.3 Finite and Infinite Set, , The order in which the elements are listed is, immaterial., , A set which has finite number of elements is called a finite, , For example, each of the following sets denotes, the same set {1, 2, 3}, {3, 2, 1}, {1, 3, 2}, , set. Otherwise, it is called an infinite set., For example, the set of all days in a week is a finite set, whereas the set of all integers, denoted by, {............ -2, -1, 0, 1, 2,...} or {x | x is an integer}, is an infinite set., An empty set I which has no element in a finite set A is, called empty of void or null set.
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SETS, RELATIONS & FUNCTIONS, , 3.4 Cardinal Number, , 4. OPERATIONS ON SETS, , The number of elements in finite set is represented by n(A),, known as Cardinal number., , 4.1 Union of Two Sets, , 3.5 Equal Sets, , The union of two sets A and B, written as A B (read as 'A, , union B'), is the set consisting of all the elements which are, , Two sets A and B are said to be equals, written as A = B, if, every element of A is in B and every element of B is in A., , either in A or in B or in both Thus,, , 3.6 Equivalent Sets, , A B = {x : x A or x B}, , Two finite sets A and B are said to be equivalent, if n, (A) = n (B). Clearly, equal sets are equivalent but equivalent, sets need not be equal., , Clearly, x A B x A or x B, and, x A B x A and x B., , For example, the sets A = { 4, 5, 3, 2} and B = {1, 6, 8, 9} are, equivalent but are not equal., 3.7 Subset, Let A and B be two sets. If every elements of A is an element, of B, then A is called a subset of B and we write A B or, B A (read as 'A is contained in B' or B contains A'). B is, called superset of A., For example, if A = {a, b, c, d} and B = {c, d, e, f}, then, A B = {a, b, c, d, e, f}, , 4.2 Intersection of Two sets, (i), , Every set is a subset and a superset itself., , (ii), , If A is not a subset of B, we write A B., , (iii), , The empty set is the subset of every set., , common elements of A and B. Thus,, , (iv), , If A is a set with n(A) = m, then the number of, subsets of A are 2m and the number of proper, subsets of A are 2m -1., , A B = {x : x A and x B}, , For example, let A = {3, 4}, then the subsets of A, , x A B x A or x B., , The intersection of two sets A and B, written as A B, (read as ‘A’ intersection ‘B’) is the set consisting of all the, , Clearly, x A B x A and x B, and, , are I , {3}, {4}. {3, 4}. Here, n(A) = 2 and number, of subsets of A = 22 = 4. Also, {3} {3,4}and {2,3}, {3, 4}, 3.8 Power Set, The set of all subsets of a given set A is called the power set, of A and is denoted by P(A)., For example, if A = {1, 2, 3}, then, P(A) = { I , {1}, {2}, {3}, {1,2} {1, 3}, {2, 3}, {1, 2, 3}}, Clearly, if A has n elements, then its power set P (A) contains, exactly 2n elements., , For example, if A = {a, b, c, d) and B = {c, d, e, f}, then, A B = {c, d}.
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SETS, RELATIONS & FUNCTIONS, 4.3 Disjoint Sets, , 4.6 Complement of a Set, , Two sets A and B are said to be disjoint, if A B = I , i.e. A, , If U is a universal set and A is a subset of U, then the, , and B have no element in common., , complement of A is the set which contains those elements, of U, which are not contained in A and is denoted by, A'or Ac. Thus,, Ac = {x : x U and x A}, , For example, if U = {1,2,3,4 ...} and A {2,4,6,8,...}, then, Ac =, {1,3,5,7, ...}, Important Results, , For example, if A = {1, 3, 5} and B = {2, 4, 6},, then A B = I , so A and B are disjoint sets., , a) Uc = I, , 4.4 Difference of Two Sets, , b) I c = U, , c) A Ac = U, , d) A Ac = I, , If A and B are two sets, then their difference A - B is defined, as :, , 5. ALGEBRA OF SETS, , A - B = {x : x A and x B}., Similarly, B - A = {x : x B and x A }., , 1., , For any set A , we have, a) A A = A, , 2., , 3., , b) A A = A, , For any set A, we have, c) A I = A, , d) A I = I, , e) A U = U, , f) A U = A, , For any two sets A and B, we have, g) A B = B A h) A B = B A, , For example, if A = {1, 2, 3, 4, 5} and B = {1, 3, 5, 7, 9} then A, - B = {2, 4} and B - A = {7, 9}., , 4., , i) A (B C) = (A B) C, , Important Results, (a), , A- B z B –A, , (b), , The sets A - B , B - A and A B are disjoint sets, , (c), , A - B A and B - A B, , (d), , A - I = A and A - A = I, , 4.5 Symmetric Difference of Two Sets, The symmetric difference of two sets A and B , denoted by, A ' B, is defined as, A ' B = (A - B) (B - A)., For example, if A = {1,2,3,4,5} and B = {1, 3,5,7,9} then, A ' B = (A - B) (B - A) = {2,4} {7,9} = {2,4,7,9}., , For any three sets A, B and C, we have, , j) A (B C) = (A B) C, 5., , For any three sets A, B and C, we have, k) A (B C) = (A B) (A C), l) A (B C) = (A B) (A C), , 6., , If A is any set, we have (Ac)c = A., , 7., , Demorgan's Laws For any three sets A, B and C, we have, m) (A B)c = Ac Bc, n) (A B)c = Ac Bc, o) A - (B C) = (A - B) ( A - C), p) A - (B C) = (A - B) (A - C)
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SETS, RELATIONS & FUNCTIONS, , Important Results on Operations on Sets, , Example – 3, , (i) A A B, B A B, A B A, A B B, (ii) A - B = A Bc, , Write the set {x : x is a positive integer and x2 < 30} in the, roster form., , (iii) (A - B) B = A B, , (iv) (A - B) B = I (v) A B Bc Ac, , Sol. The squares of positive integers whose squares are less, than 30 are : 1, 2, 3, 4, 5., , (vi) A - B = Bc - Ac, , (vii) (A B) (A Bc) = A, , (viii) A B = (A - B) (B - A) (A B), , Hence the given set, in roster form, is {1, 2, 3, 4, 5}., Example – 4, , (ix) A - (A - B) = A B, (x)A - B = B -A A= B, , Write the set {0, 1, 4, 9, 16, .......} in set builder form., , (xi)A B =A B A= B, , (xii) A (B ' C) = (A B) ' (A C), , Sol. The elements of the given set are squares of integers :, , 0, r 1, r 2, r 3, r 4, ......., , Example – 1, , Hence the given set, in set builder form, is {x2 : xZ}., Write the set of all positive integers whose cube is odd., Example – 5, Sol. The elements of the required set are not even., , , , State which of the following sets are finite and which are, , [' Cube of an even integer is also an even integer], , infinite, , Moreover, the cube of a positive odd integer is a positive, , (i) A = {x : x N and x2 – 3x + 2 = 0}, , odd integer., , (ii) B = {x : x N and x2 = 9}, , The elements of the required set are all positive odd integers., , (iii) C = {x : x N and x is even}, , Hence, the required set, in the set builder form, is :, , (iv) D = {x : x N and 2x – 3 = 0}., , 2k 1 : k t 0, k Z ., Sol. (i) A = {1, 2}., [' x2 – 3x + 2 = 0 (x – 1) (x – 2) = 0 x = 1, 2], , Example – 2, , Hence A is finite., , 1 2 3 4 5 6 7½, Write the set ® , , , , , , ¾ in the set, ¯2 3 4 5 6 7 8¿, builder form., Sol. In each element of the given set the denominator is one, more than the numerator., Also the numerators are from 1 to 7., Hence the set builder form of the given set is :, , x:x, , (ii), , B = {3}., [' x2 = 9 x = + 3. But 3 N], Hence B is finite., , (iii) C = {2, 4, 6, ......}, Hence C is infinite., , ª, (iv) D = I. «' 2x 3 0 x, ¬, , n / n 1, n N and 1 d n d 7 ., Hence D is finite., , 3, º, N», 2, ¼
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SETS, RELATIONS & FUNCTIONS, Example – 6, , Example – 8, , Which of the following are empty (null) sets ?, , Are the following pairs of sets equal ? Give reasons., , (i) Set of odd natural numbers divisible by 2, , (i) A = {1, 2}, B = {x : x is a solution of x2 + 3x + 2 = 0}, , (ii) {x : 3 < x < 4, x N}, , (ii) A = {x : x is a letter in the word FOLLOW},, , (iii) {x : x2 = 25 and x is an odd integer}, , B = {y : y is a letter in the word WOLF}., , (iv) [x : x2 – 2 = 0 and x is rational], (v) {x : x is common point of any two parallel lines}., , Sol. (i) A = {1, 2}, B = {–2, –1}, [' x2 + 3x + 2 = 0 (x + 2) (x + 1) = 0 x = –2, —1], , Sol. (i) Since there is no odd natural number, which is divisible, by 2., , Clearly A z B., , ? it is an empty set., (ii), , Since there is no natural number between 3 and 4., , (ii), , B = {W, O, L, F} = {F, O, L, W}., , ? it is an empty set., (iii) Now x2 = 25 x = + 5, both are odd., ? The set {– 5, 5} is non-emptry., , A = {F, O, L, L, O, W} = {F, O, L, W}, , Clearly A = B., Example – 9, , (iv) Since there is no rational number whose square is 2,, , (v), , ? the given set is an empty set., , Let A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, C = {6, 7, 8, 9} and, , Since any two parallel lines have no common point,, , D = {7, 8, 9, 10}. Find :, , ? the given set is an empty set., , (a), , Example – 7, Find the pairs of equal sets from the following sets, if any,, giving reasons :, , (b), , (i) A B, , (ii) B D, , (iii) A B C, , (iv) B C D, , (i) A B, , (ii) B D, , (iii) A B C., , A = {0}, B = {x : x > 15 and x < 5},, C = {x : x – 5 = 0}, D = {x : x2 = 25},, E = {x : x is a positive integral root of the equation, x2 – 2x – 15 = 0}., , Sol. (a) (i) A B = {1, 2, 3, 4, 5} {3, 4, 5, 6, 7}, = {1, 2, 3, 4, 5, 6, 7}., (ii), , = {3, 4, 5, 6, 7, 8, 9, 10}., , Sol. Here we have,, A = {0}, , (iii), , B=I, [' There is no number, which is greater than 15 and less, than 5], , C = {5}, , and E = {5}., [' x2 – 2x – 15 = 0 (x – 5) (x + 3) = 0 x = 5, – 3. Out of, these two,, 5 is positive integral], Clearly C = E., , A B C = {1, 2, 3, 4, 5} {3, 4, 5, 6, 7} {6, 7, 8, 9}., = {1, 2, 3, 4, 5, 6, 7} {6, 7, 8, 9} = {1, 2, 3, 4, 5, 6, 7, 8, 9}., , (iv), , [' x – 5 = 0 x = 5], , D = {– 5, 5} [' x2 = 25 x = + 5], , B D = {3, 4, 5, 6, 7} {7, 8, 9, 10}, , B C D = {3, 4, 5, 6, 7} {6, 7, 8, 9} {7, 8, 9, 10}., = {3, 4, 5, 6, 7, 8, 9} {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}., , (b) (i) A B = {1, 2, 3, 4, 5} {3, 4, 5, 6, 7} = {3, 4, 5}., (ii), , B D = {3, 4, 5, 6, 7} {7, 8, 9, 10} = {7}., , (iii), , A B C = {1, 2, 3, 4, 5} {3, 4, 5, 6, 7} {6, 7, 8, 9} = {3,, 4, 5} {6, 7, 8, 9} = I.
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SETS, RELATIONS & FUNCTIONS, Example – 13, , Example – 10, , Let A = {1, 2, 3, 4, 5, 6}, B = {3, 4, 5, 6, 7, 8}. Find, (A – B) (B – A)., , If A1 = {2, 3, 4, 5}, A2 = {3, 4, 5, 6}, A3 = {4, 5, 6, 7}, find, , Ai and Ai, where i = {1, 2, 3}., , Sol. We have, A = {1, 2, 3, 4, 5, 6} and B = {3, 4, 5, 6, 7, 8}., Sol. (i), , (ii), , A i = A1 A 2 A 3 = {2, 3, 4, 5} {3, 4, 5, 6} , , ? A – B = {1, 2} and B – A = {7, 8}, , {4, 5, 6, 7}, , ?(A – B) (B – A) = {1, 2} {7, 8} = {1, 2, 7, 8}., , = {2, 3, 4, 5} {3, 4, 5, 6, 7} = {2, 3, 4, 5, 6, 7}., , Some Basis Results about Cardinal Number, , A i = A1 A 2 A 3 = {2, 3, 4, 5} {3, 4, 5, 6} , , If A, B and C are finite sets and U be the finite universal set,, then, , {4, 5, 6, 7}, = {2, 3, 4, 5} {4, 5, 6} = {4, 5}., Example – 11, Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4},, B = {2, 4, 6, 8}. Find :, , (i), , n (Ac) = n (U) - n (A), , (ii), , n (A B) = n (A) + n (B) - n (A B), , (iii) n (A B) = n(A) + n (B), where A and B are disjoint non empty sets., (iv) n (A Bc) = n (A) - n (A B), (v), , (i) AC (ii) BC, , (iii) (AC)C, , (iv) A B, , C, , n (Ac Bc) = n (A B)c = n (U) - n (A B), , (vi) n (Ac Bc) = n (A B)c = n (U) - n (A B), (vii) n (A - B) = n (A) - n (A B), , Sol. (i), (ii), , A = Set of those elements of U, which are not in, , (viii) n (A B) = n (A B) - n (A Bc) - n (Ac B), , A = {5, 6, 7, 8, 9}., , (ix), , n (A B C) = n (A) + n (B) + n (C) - n (A B) – n(B C) –, n(C A) + n (A B C), , (x), , If A1, A2, A3, ... An are disjoint sets , then, , C, , C, , B = Set of those elements of U, which are not in, B = {1, 3, 5, 7, 9}., , (iii), , n (A1 A2 A3 ... An) = n(A1) + n (A2) + n (A3), + ... + n(An), , (AC)C = Set of those elements of U, which are not in, A’ = {1, 2, 3, 4} = A., , (iv), , A B = {1, 2, 3, 4} {2, 4, 6, 8} = {1, 2, 3, 4, 6, 8}., , AB, , C, , = Set of those elements of U, which are not in, , (xi), , Example – 14, , A B = {5, 7, 9}., , If A = {1, 2, 3}, B = {4, 5, 6} and C = {7, 8, 9}, verify that, , A B C = A B A C ., , Example – 12, If U = {x : x is a letter in English alphabet},, , Sol. We have, A = {1, 2, 3}, B = {4, 5, 6} and C = {7, 8, 9}., , A = {x : x is a vowel in English alphabet}., , A B = {1, 2, 3} {4, 5, 6} = {1, 2, 3, 4, 5, 6} ...(1), , Find AC and (AC)C., , A C = {1, 2, 3} {7, 8, 9}, , Sol. (i) Since A = {x : x is a letter in English alphabet},, ? AC is the set of those elements of U, which are not vowels, , = {1, 2, 3, 7, 8, 9}, and B C = {4, 5, 6} {7, 8, 9} = I, , ...(2), ...(3), , Now A B C, , ...(4), , = {x : x is a consonant in English alphabet}., (ii), , n (A ' B) = number of elements which belong to exactly, one of A or B., , (AC)C is the set of those elements of U, which are not, , and, , 1, 2, 3 I = {1, 2, 3}, , A B A C = {1, 2, 3, 4, 5, 6} {1, 2, 3, 7, 8, 9}, , consonants = {x : x is a vowel, , = {1, 2, 3}, , in English alphabet} = A., , From (4) and (5), A B C, , C C, , Hence (A ) = A., , verifies the result., , ...(5), , A B A C , which
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SETS, RELATIONS & FUNCTIONS, Example – 17, , Example – 15, Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and, B = {2, 3, 5, 7}. Verify that, (i) A B, , C, , (ii) A B, , A C BC, , C, , A C BC ., , Sol. We have, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}., (i), , A B = {2, 4, 6, 8} {2, 3, 5, 7}, = {2, 3, 4, 5, 6, 7, 8}, , A B, , C, , = {1, 9}, , ...(1), , Also AC = {1, 3, 5, 7, 9}, and BC = {1, 4, 6, 8, 9}, , A C BC = {1, 3, 5, 7, 9} {1, 4, 6, 8, 9}, = {1, 9}, , ...(2), , From (1) and (2), A B, , C, , A C BC , which verifies the, , result., (ii), , A B = {2, 4, 6, 8} {2, 3, 5, 7} = {2}, A B, , C, , = {1, 3, 4, 5, 6, 7, 8, 9}, , ...(3), , Prove that :, A (B – C) = (A B) – (A C), Sol. Let x be an arbitrary element of A (B – C)., Then x A (B – C), , x A and x (B – C), , x A and (x B and x C), , (x A and x B) and (x A and x C), , x (A B) and x (A C), , x {(A B) – (A C)}, A (B – C) (A B) – (A C), ... (1), Let y be an arbitrary element of (A B) – (A C)., Then y (A B) – (A C), , y (A B) and y (A C), , (y A and y B) and (y A and y C), , y A and (y B and y C), , y A and y (B – C), , y A (B – C), (A B) – (A C) A (B – C), ... (2), Combining (1) and (2)., A (B – C) = (A B) – (A C)., Example – 18, , and A C BC = {1, 3, 5, 7, 9} {1, 4, 6, 8, 9}, Prove the following :, , = {1, 3, 4, 5, 6, 7, 8, 9} ...(4), From (3) and (4), A B, , C, , A C BC , which verifies the, , result., , Sol. Let x Bc, where x is arbitrary., , Example – 16, , Sol., , A B Bc Ac, , Now x Bc, , If A and B are any two sets, prove using Venn Diagrams, , , , xB, , (i) A – B = A BC, , , , x A, , , , x Ac, , (ii) (A – B) B = A B., , Bc Ac, (i), , [' A B], , ... (1), , Conversely : Let x A, where x is arbitrary., Now x A, , (ii), , , , x Ac, , , , x Bc, , , , x B, , [' Bc Ac], , A B, Combining (1) and (2), A B Bc Ac.
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SETS, RELATIONS & FUNCTIONS, n(I M) = 15, n(M A I) = 5, , Example – 19, , n(X) = 200, , Prove the following :, , n(M A I) = n(M) + n(A) + n (I) –, , A – B = A – (A B), , n(M A) –n (A I) – n (M I) + n (M A I), , where U is the universal set., , = 35 + 40 + 40 – 20 – 17 – 15 + 5 = 68, , Sol. Let x (A – B), where x is arbitrary., , (i), , Now x (A – B), , Number of students passed in all three examination, = 200 – 68 = 132, , , , x A and x B, , , , (x A and x A) and x B, , =n (I A) = n(I) + n(A) – n (I A), , [Note this step], , = 40 + 40 – 17 = 63, , , , x A and (x A and x B), , (ii), , Number of students failed in IIT or AIEEE, , Example – 21, , [Associative Law], , , x A and x (A B), , , , x A – (A B), , Hence, , In a hostel, 25 students take tea, 20 students take coffee,, 15 students take milk, 10 students take both tea and coffee,, 8 students take both milk and coffee. None of the them, , A – B = A – (A B)., , take tea and milk both and everyone takes atleast one, beverage, find the number of students in the hostel., , Example – 20, In a class of 200 students who appeared in a certain, examination. 35 students failed in MHTCET, 40 in AIEEE,, 40 in IIT, 20 failed in MHTCET and AIEEE, 17 in AIEEE, , Sol., , and IIT, 15 in MHTCET and IIT and 5 failed in all three, examinations. Find how many students, (i), , Did not fail in any examination., , (ii), , Failed in AIEEE or IIT., , Let the sets, T and C and set M are the students who drink, tea, coffee and milk respectively. This problem can be solved, by Venn diagram., n(T) = 25; n(C) = 20; n (M) = 15, n(T C) = 10; n(M C) = 8, , Sol., , Number of students in hostel, = n (T C M), ?n(T C M) = 15 + 10 + 2 + 8 + 7 = 42, , n(M) = 35, n(A) = 40, n(I) = 40, n(M A) = 20, n(A I) = 17,
