1 Sets The set theory was developed by German Mathematician Georg Cantor (1845-1918). He first encountered sets while working on problems on trigonometric series. This concept is used in every branch of Mathematics i.e., Geometry, lgebra, etc. 1 Sets and Their Types In our daily life, while performing our regular work, we often come across a variety of things that occur in groups. e.g., Team of cricket players, group of tall boys, group of teachers, etc. The words used above like team, group, etc., convey the idea of certain collections. Well-defid Collection of Objects If any given collection of objects is in such a way that it is possible to tell without any doubt whether a given object belongs to this collection or not, then such a collection of objects is called a well-defid collection of objects. e.g., The rivers of India is a well-defid collection. Since, we can say that the river Nile does not belong to this collection. On the other hand, the river Ganga does belong to this collection. Difference between Not Well-defid Collection and Well-defid Collection Sets and Their Types Subsets of Set Venn Diagrams and Operations on Sets pplications of Set Theory Not Well-defid Collection group of intelligent students. group of most talented writers of India. Group of pretty girls. Well-defid Collection group of students scoring more than 95% marks of your school. group of odd natural numbers less than 5. Group of girls of class XI of your school. Here, a group of intelligent students, a group of most talented writers of India, group of pretty girls are not well-defid collections, because we can not decide whether a given particular object belongs to the given collection or not. 1
ll Mathematics Class 11th Definition of Set well-defid collection of objects, is called a set. Sets are usually denoted by the capital letters,, C, X, Y and Z etc. The elements of a set are represented by small letters a, b, c, x, y and z etc. If a is an element of a set, then we say that a belongs to. The word belongs to denoted by the Greek symbol (epsilon). Thus, in notation form, a belongs to set is written as a and b does not belongs to set is written as b. e.g., (i) 6 N, N being the set of natural numbers and0 N. (ii) 36 P, P being the set of perfect square numbers, so 5 P. * Some examples of sets used particularly in Mathematics are N The set of natural numbers. {1,, 3, } Z The set of all integers. {..., 3,, 1, 0, 1,, 3,... } Q The set of all rational numbers. 3 1 1 3...,,,,,,,... 4 3 3 4 R The set of real numbers. (rational and irrational numbers) Z + The set of positive integers. {1,, 3, } Q + The set of positive rational numbers. 1 3 4, 3, 4, 5,... R + The set of positive real numbers. (positive rational and irrational numbers) Note Objects, elements and members of a set are synonymous terms. Example 1. Which of the following are sets? Justify your answer. (i) The collection of all the months of a year beginning with the letter J. (ii) The collection of ten most talented writers of India. (iii) collection of novels written by the writer Munshi Prem Chand. (iv) collection of most dangerous animals of the world. Solution (i) We are sure that members of this collection are January, Ju and July. So, this collection is well-defid. Hence, it is a set. (ii) writer may be most talented for o person and may not be for other. Therefore, we cannot definitely decide which writer will be there in the collection. So, this collection is not well-defid. Hence, it is not a set. (iii) Here, we can definitely decide whether a given novel belongs to this collection or not. So, this collection is well-defid. Hence, it is a set. (iv) The term most dangerous is not a well-defid term. n animal may be most dangerous for o person and may not be for the other. So, it is not well-defid. Hence, it is not a set. Representation of Sets Sets are gerally represented by following two ways 1. Roster or Tabular Form or Listing Method. Set-builder Form or Rule Method 1. Roster or Tabular Form or Listing Method In this form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within curly braces { }. e.g., (i) The set of all natural numbers less than 10 is represented in roster form as {1,, 3, 4, 5, 6, 7, 8, 9}. (ii) The set of prime natural numbers is {, 3, 5, 7, K }. Here, three dots tell us that the list of prime natural numbers continue indefinitely. Note (i) In roster form, order in which the elements are listed is not important. Hence, the set of natural numbers less than 10 can also be written as {, 4, 1, 3, 5, 6, 8, 7, 9} instead of {1,, 3, 4, 5, 6, 7, 8, 9}. Here, the order of listing elements is not important. (ii) In roster form, element is not repeated i.e., all the elements are taken as distinct. Hence, the set of letters forming the word MISCELLNEOS is {M, I, S, C, E, L,, N, O, }.. Set-builder Form or Rule Method In this form, all the elements of set possess a single common property p( x), which is not possessed by any other element outside the set. In such a case, the set is described by { x : p( x) holds}. e.g., In the set {a, e, i, o, u}, all the elements possess a common property, i.e., each of them is a vowel of the English alphabet and no other letter possesses this property. If we denote the set of vowels by V, then we write V = {x : x is a vowel in the English alphabet} The above description of the set V is read as the set of all x such that x is a vowel of the English alphabet.
Sets 3 Some examples are given below (i) Set of all natural numbers less than 10, = { x : x N, x < 10} (ii) Set of all real numbers, R = { x : x R} * Working Rule to Write the Set in the Set-builder Form To convert the given set in set-builder form, we use the following steps Step I Describe the elements of the set by using a symbol x or any other symbol y, z etc. Step II Write the symbol colon :. Step III fter the sign of colon, write the characteristic property possessed by the elements of the set. Step IV Enclose the whole description within braces i.e., { }. out Problem Write the following set = {14, 1, 8, 35, 4,..., 98} in set-builder form. Step I Describe the elements of the set by using a symbol. Let x represent the elements of given set. Step II Write the symbol colon. Write the symbol : after x. Step III Find the characteristic property possessed by the elements of the set. Given numbers are all natural numbers greater than 7 which are multiples of 7 and less than 100. Step IV Enclose the whole description within braces. Thus, = {x : x is a set of natural numbers greater than 7 which are multiples of 7 and less than 100}. Which is the required set- builder form of given set. Representation of a Statement in oth Form Statement Roster form Set- builder form The set of all natural numbers between 10 and 14. The set of all schools in Delhi beginning with letter D. The set of all distinct letters used in the word Friend. {11, 1, 13} { x : x N, 10 < x < 14} {DV School, DPS School, Decent School,...} {x : x is name of school in Delhi beginning with letter D} {F, r, i, e, n, d} {x : x is the distinct letters used in the word Friend } Example. Describe the set (i) The set of all vowels in the word EQTION in Roster form. (ii) The set of reciprocals of natural numbers in set-builder form. Solution (i) The word EQTION has following vowels i.e.,, E, I, O and. Hence, the required set can be described in roster form as {, E, I, O, }. (ii) Given set can be described in set-builder form as { x : x is a reciprocal of a natural number} or x : x = 1 n, n N Example 3. Describe the following set in Roster form. {x : x is positive integer and a divisor of 9} Solution Here, x is a positive integer and a divisor of 9. So, x can take values 1, 3, 9. {x : x is a positive integer and a divisor of 9} = { 1, 3, 9} Example 4. Write the set of all natural numbers x such that 4x + 9 < 50 in roster form. Firstly, simplify the iquality and then listed all the natural numbers under given condition. Solution We have, 4x + 9 < 50 4x + 9 9 < 50 9 [subtracting 9 from both sides] 4x < 41 x < 41 4 x < 10.5 Since, x is a natural number, so x can take values 1,, 3, 4, 5, 6, 7, 8, 9, 10. Required set = {1,, 3, 4, 5, 6, 7, 8, 9, 10} Example 5. Describe the following set in set-builder form. = { 53, 59, 61, 67, 71, 73, 79, 83, 89, 97} Firstly, find the characteristic property possessed by the elements of the set, then write in set-builder form. Solution The given set is {53, 59, 61, 67, 71, 73, 79, 83, 89, 97}. We observe that these numbers are all prime numbers between 50 and 100. Given set, = { x : x is a prime number and 50 < x < 100} 3
4 ll Mathematics Class 11th Types of Sets 1. Finite Set set, which is empty or consists of a definite number of elements, is called a finite set. e.g., (i) The set {1,, 3, 4} is a finite set, because it contains a definite number of elements i.e., only 4 elements. (ii) The set of solutions of x = 5 is a finite set, because it contains a definite number of elements i.e., 5 and 5. (iii) n empty set, which does not contain any element is also a finite set. The number of distinct elements in a finite set is called cardinal number of set and it is denoted by n ( ). e.g., If = { 3, 1, 8, 10, 13 }, then n ( ) = 5.. Infinite Set set which consists of infinite number of elements is called an infinite set. When we represent an infinite set in the roster form, it is not possible to write all the elements within braces { } because the number of elements of such a set is not finite, so we write a few elements which clearly indicate the structure of the set following by three dots. e.g., Set of squares of natural numbers is an infinite set, because such natural numbers are infinite and it can be represented as { 4, 9, 16, 5...}. Note ll infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern. 3. Empty Set set, which does not contain any element, is called an empty set or null set or void set. It is denoted by φ or { }. e.g., = { x : x is a natural number less than 1} We know that, there is no natural number less than o. Therefore, set contains no element and hence it is an empty set. 4. Singleton Set set, consisting of a single element, is called a singleton set. e.g., (i) The sets {0}, {5}, { 7} are singleton sets. (ii) = { x : x + 8 = 0, x Z } is a singleton set, because this set contains only o integer, namely 8. 5. Equivalent Sets Two sets and are equivalent, if their cardinal numbers are same i.e., n ( ) = n( ). e.g., Let = { a, b, c, d } and = { 1,, 3, 4}, then n( ) = 4 and n( ) = 4. Therefore, and are equivalent sets. 6. Equal Sets Two sets and are said to be equal, if they have exactly the same elements and we write =. Otherwise, two sets are said to be uqual and we write. e.g., Let = { a, b, c, d } and = { c, d, b, a}, then =, because each element of is in and vice-versa. Note set does not change, if o or more elements of the set are repeated. e.g., The sets = { 1, 4, 5} and = { 1, 1, 4, 5, 5} are equal because elements of is in and vice-versa. That s why, we gerally do not repeat any element in describing a set. Example 6. Identify which of the following set is an empty set, singleton set, infinite set or equal sets. (i) = { x : x is a girl being living on the Jupiter} (ii) = {x : x is a letter in the word MRS } (iii) C = { y : y is a letter in the word RMS } (iv) D = { x : 3x = 0, x Q} (v) E = { x : x N and x is an odd number} Solution (i) = { x : x is a girl being living on the Jupiter} We know that, there is no human being or any girl being living on the Jupiter. Hence, is an empty set. (ii) = { x : x is a letter in the word MRS } = {M,, R, S} (iii) C = { y : y is a letter in the word RMS } C = {, R, M, S} Here, we observe that the elements of sets and C are exactly same, hence these sets are equal. (iv) D = { x : 3x = 0, x Q} D = Q 3x = 0 x = Q 3 3 Hence, D is a singleton set. (v) E = { x : x N and x is an odd number} Clearly, it is an infinite set because there are infinite natural numbers which are odd. 4
Sets 5 Example 7. Which of the following pairs of sets are equal? Justify your answer. (i) = { x : x is a letter of the word LOYL }, = { x : x is a letter of the word LLOY } (ii) = { x : x Z and x < 8}, = { x : x R and x 4x + 3 = 0} Firstly, describe the given sets in the roster form and check whether they have exactly same elements. Solution (i) Given, = { x : x is a letter in the word LOYL } = {L, O, Y, } = {, L, O, Y} and = { x : x is a letter of the word LLOY } = {, L, O, Y} Here, we see that both sets have exactly the same elements. = (ii) Given, = { x : x Z and x < 8} = {, 1, 0, 1, } [Qx < 8 < x < and as x Z x {, 1, 0, 1, }] and = { x : x R and x 4x + 3 = 0} = { 1, 3} [Qx 4x + 3 = 0 ( x 1)( x 3) = 0 x = 1, 3] Here, we see that set has 5 distinct elements and set has distinct elements. So, they do not have same elements. EXM Very Short nswer Type Questions [1 Mark each] 1. If = { 1, 3, 5, 7, 9, 11, 13, 15 }, then insert the appropriate symbol or in each of the following blank spaces. (i) 1... (ii) 6... (iii) 9... (iv) 14... Sol. Given set is = { 1, 3, 5, 7, 9, 11, 13, 15} (i) 1 (ii) 6 (iii) 9 (iv) 14. Write {x : x is an integer and 3 x < 7} in roster form. Sol. { 3,, 1, 0, 1,, 3, 4, 5, 6 } 3. Write set = { 3, 6, 9, 1, 15} in set-builder form. Sol. = {x : x is a natural number multiple of 3 and x < 18}. 4. Write set ={1, 4, 9,..., 100} in set-builder form. Sol. = { x : x = n, n N and n < 11} 5. Write set D = 1 3 4 5 6 7,,,,,, in 5 10 17 6 37 50 set-builder form. n Sol. D = x : x =, n N and n 7 n + 1 6. se listing method to express the set = {x : x = n 3, n N and x < 80}. Sol. = { 1, 8, 7, 64 } 7. List the elements of the set 1 9 = x : x is an integer, < x < Sol. = { 0, 1,, 3, 4 } 8. Express the set n 1 D = x : x =, n N and n < 4 in roster form. n + 1 Sol. D = 0 3 4,, 5 5 9. Describe (i) The set of vowels in the word MTHEMTICS in roster form. (ii) The set of all odd natural numbers in set builder form. Sol. (i) The word MTHEMTICS has following vowels i.e.,, E, I. Hence, the required set can be described in roster form as {, E, I}. [1/] (ii) n odd natural number can be written in the form ( n 1). Hence, given set can be described in set-builder form as { x : x = n 1, n N } [1/] 5
6 ll Mathematics Class 11th Short nswer Type Questions [4 Marks each] 10. Which of the following sets are empty? (i) = {x : x N and x 1} (ii) = {x : 3x + 1 = 0, x N} (iii) C = {x : < x < 3, x N} set which does not contain any element is called the empty set. Sol. (i) = {x : x N and x 1} Since, x 1i.e., x < 1and x = 1, which is a natural number. = { 1} So, this is not an empty set. (ii) = {x : 3x + 1 = 0, x N} Since, 3x + 1 = 0 x = 1 N 3 Thus, set does not contain any element. Hence, set is an empty set. [ 1 1 ] (iii) C = {x : < x < 3, x N} Since, there is no natural number between and 3. Hence, set C is an empty set. [ 1 1 ] 11. Let T = x : x x + 5 7 4x 40 5 =. 13 x Is T an empty set? Justify your answer. Sol. The given set T is not an empty set. Justification x + 5 4x 40 Q 5 = x 7 13 x x + 5 5 4x 40 = x 7 1 13 x x + 5 5 x + 35 4x 40 = x 7 13 x 40 4x 4x 40 = x 7 13 x ( 4x 40) 4x 40 = 0 x 7 13 x 1 1 ( 4x 40) + 0 ( 13 x ) ( x 7 ) = 6 ( 4x 40) 0 ( 13 x ) ( x 7 ) = 4x 40 = 0 x = 10 Hence, T is not an empty set. 1. Which of the following sets are finite and which are infinite? (i) The set of the months of a year. (ii) {1,, 3,...} (iii) The set of prime numbers less than 99. Sol. (i) It is a finite set, as there are 1 members in the set which are months of the year. (ii) It is an infinite set, since there are infinite natural numbers. [ 1 1 ] (iii) It is a finite set, because the set is {, 3, 5, 7,..., 97}. [ 1 1 ] 13. From the following sets, select equal sets. = {, 4, 6, 8 }, = { 1,, 3, 4, 5 }, C = {, 4, 6, 8 }, D = {, 3, 5, 4, 1 }, E = { 8, 6,, 4} Two sets are said to be equal, if they have exactly the same elements. Sol. = {, 4, 6, 8} and E = { 8, 6,, 4} Since, each element of set presents in set E and vice-versa. Therefore, and E are equal sets. Similarly, = { 1,, 3, 4, 5} and D = {, 3, 5, 4, 1} Since, each element of set presents in set D and vice-versa. Therefore, and D are also equal sets. 14. re the following pair of sets equal? Give reason. (i) = {, 3} and = {x : x is a solution of x + 5x + 6 = 0} (ii) = {x : x is a letter in the word FOLLOW } and = {y : y is a letter in the word WOLF } Firstly, convert the given sets in roster form and then check whether they have exactly the same elements. Sol. (i) Here, = {, 3} and = {x : x is a solution of x + 5x + 6 = 0} First, we find the solution of x + 5x + 6 = 0. Now, x + 3x + x + 6 = 0 x( x + 3) + ( x + 3) = 0 ( x + )( x + 3) = 0 x =, 3 = {, 3 } Since, the elements of and are not same, therefore. (ii) Here, = {x : x is a letter of the word FOLLOW } = {F, O, L, W} and = { y : y is a letter of the word WOLF } = {W, O, L, F} Since, every element of is in and every element of is in i.e., both have exactly same elements. = 6
TRY YORSELF 1. Write set = {x : x is an integer and x < 0} in the roster form.. Write set = {x : x is a plat} in the roster form. 3. Which of the following is/are the examples of an empty set? (i) Set of all even prime numbers. (ii) {x : x is a point common to any two parallel lis} 4. Write the following sets in the roster form. (i) = { x : x = x, x R} (ii) C = {x : x is a positive factor of a prime number p} 5. Write the following sets in the set-builder form. (i) { 5, 5, 15, 65} (ii) {, 4, 6, 8, 10} 6. From the sets given below, pair the equivalent sets. = { 1,, 3}, = { t, p, q, r, s}, C = { α, β, γ} and D = { a, e, i, o, u} 7. From the sets given below, select empty set, singleton set, infinite set and equal sets. 3 (i) = {x : x < 1and x > 3} (ii) = { x : x 1= 0, x R} (iii) C = {x : x N and x is a prime number} (iv) D = {, 4, 6, 8, 10} (v) E = {x : x is positive even integer and x 10} 8. From the sets given below, select equal sets and equivalent sets. = { 0, a}, = { 1,, 3, 4 }, C = { 4, 8, 1 }, D = { 3, 1,, 4 }, E = { 1, 0 }, F = { 8, 4, 1 }, G = { 1, 5, 7, 11} and H = { a, b} 9. State which of the following statements are true and which are false? Justify your answer. (i) 35 {x : x has exactly four positive factors} (ii) 18 {y :the sum of all the positive factors of y is y} 4 3 (iii) 3 { x : x 5x + x 11x + 6 = 0} (iv) 496 {y :the sum of all the positive factors of y is y} 7
Subset Let and be two sets. If every element of is an element of, then is called a subset of. If is a subset of, then we write, which is read as is a subset of or is contaid in. In other words,, if whever a, then a. It is often convenient to use the symbol which means implies. sing this symbol, we can write the definition of subset as follows, if x x The above statement is read as is subset of, if x is an element of, then it implies that x is also an element of. If is not a subset of, then we write. If, then is called a superset of written as. e.g., Consider the sets and, where set denotes the set of all students in your class, denotes the set of all students in your school. We observe that, every element of is also an element of. Therefore, we can say that is subset of i.e.,. If it happens for both sets and, i.e., every element of is in and every element of is in, then in this case, and are same sets. Thus we have, and =, where is a symbol for two ways implications and is usually read as if and only if (iff ). Note (i) { 1} { 1,, 3} (ii) { 1,, 3}, which is not possible. Proper Subset Subsets of Set If and, then is called a proper subset of, written as and is called proper superset of. e.g., Let = {x : x is an even natural number} and = { x : x is a natural number} Then, = {, 4, 6, 8,...} and = {1,, 3, 4, 5,...} Some Important Results 1. Every set is a subset of itself. Proof Let be any set. Then, each element of is clearly in. Hence,.. The empty set φis a subset of every set. Proof Let be any set and φbe the empty set. To show that φ, i.e., to show that every element of φis an element of also. ut we know that empty set φcontains no element. So, every element of φis in. Hence, φ. 3. set itself and an empty set are always subsets of every set and set itself is called improper subset of the set. 4. The total number of subsets and proper subset of a finite set containing n elements is n and n 1, respectively. 5. If and C, then C. 6. =, if and only if and. Example 1. Let = {1,, {3, 4}, 5}. Which of the following statements are incorrect and why? (i) {3, 4} (ii) {3, 4} (iii) {{ 3, 4}} (iv) 1 (v) 1 (vi) {1,, 5} (vii) {1,, 5} (viii) φ (x) { φ} Solution We have, = {1,, {3, 4}, 5} (ix) φ (i) Since, {3, 4} is a member of set. {3, 4} Hence, {3, 4} is incorrect. (ii) Since, {3, 4} is a member of set. Hence, {3, 4} is correct. (iii) Since, {3, 4} is a member of set. So, {{ 3, 4}} is a subset of. Hence, {{ 3, 4}} is correct. (iv) Since, 1 is a member of. Hence, 1 is correct. (v) Since, 1 is a member of set. Hence, 1 is incorrect. (vi) Since, 1,, 5 are members of set. So, {1,, 5} is a subset of set. Hence, {1,, 5} is correct. (vii) Since, 1, and 5 are members of set. So, {1,, 5} is a subset of. Hence, {1,, 5} is incorrect. (viii) Since, φ is subset of every set. Hence, φ is correct. (ix) Since, φ is not a member of set. Hence, φ is incorrect. (x) Since, φ is not a member of set. Hence, { φ} is incorrect. 8
Sets 9 Subsets of the Set of Real Numbers Rational numbers and irrational numbers taken together, are known as real numbers. Thus, every real number is either a rational or an irrational number. The set of real numbers is denoted by R. There are many important subsets of set of real numbers which are given below 1. Natural Numbers The numbers being used in counting as 1,, 3, 4,..., called natural numbers. The set of natural numbers is denoted by N. N = {1,, 3, 4, 5, 6,...}. Whole Numbers The natural numbers along with number 0 (zero) form the set of whole numbers i.e., 0, 1,, 3,..., are whole numbers. The set of whole numbers is denoted by W. W = {0, 1,, 3,...} Set of natural numbers is the proper subset of the set of whole numbers. N W 3. Integers The natural numbers, their gatives and zero make the set of integers and it is denoted by Z. Z = {..., 5, 4, 3,, 1, 0, 1,, 3, 4,...} Set of whole numbers is the proper subset of integers. W Z 4. Rational Numbers number of the form p, where p and q both are integers q and q 0, is called a rational number (division by 0 is not permissible). The set of rational numbers is gerally denoted by Q. p Q = : p, q Z and q 0 q ll the whole numbers are also rational numbers, since they can be represented as the ratio. 3 6 18 e.g., 3 =,,, etc. 1 6 The set of integers is the proper subset of the set of rational numbers i.e., Z Q and N Z Q. 5. Irrational Numbers number which cannot be written in the form p/q, where p and q both are integers and q 0, is called an irrational number i.e., a number which is not rational is called an irrational number. The set of irrational numbers is denoted by T. T = { x: x R and x Q } e.g., 0.535335333...,, 3 are irrational numbers. bove subsets can be represented diagramatically as given below Non-gative rational number Whole number Natural number Real number Intervals as Subsets of R Let a, b belongs to R and a < b. Then, the set of real numbers { x : a < x < b} is called an open interval and is denoted by ( a, b ). ll the real numbers between a and b belongs to the open interval (a, b) but a and b do not belong to this set (interval). The interval which contains the first and last members of the set, is called closed interval and it is denoted by [a, b]. Here, [ a, b] = { x : a x b} Semi-open or Semi-closed Interval Some intervals are closed at o end and open at the other, such intervals are called semi-closed or semi-open intervals. [ a, b) = { x : a x < b} is an open interval from a to b which includes a but excludes b. ( a, b] = { x : a < x b} is an open interval from a to b which excludes a but includes b. e.g., (i) (, 8) is a subset of ( 1, 11 ). (ii) [4, 6) is a subset of [4, 6]. (iii) The set [ 0, ) defis the set of non-gative real numbers. (iv) The set (, 0) defis the set of gative real numbers. (v) (, ) is the set of real numbers. Negative rational number Irrational number 9
10 ll Mathematics Class 11th On real li, we can draw the interval, which is shown by the dark portion on the number li a a ( a, b) [ a, b) Length of an interval The number ( b a) is called the length of any of the intervals ( a, b ),[ a, b ],[ a, b) or ( a, b ]. Example. (a) Write the following as intervals (i) { x : x R, 5 < x 6 } (ii) { x : x R, 11 < x < 9} (b) Write the following as intervals and also represent on the number li. (i){ x : x R, x < 8 } (ii) { x : x R, 5 x 6} If an iquality is of the form or, then we use the symbol of closed interval, otherwise we use the symbol of open interval. Solution (a) (i) { x : x R, 5 < x 6} is the set that does not contain 5 but contains 6. So, it can be written as an interval whose first end point is open and last end point is closed i.e., ( 5, 6 ]. (ii) { x : x R, 11< x < 9} is the set that ither contains 11nor 9, so it can be represented as open interval i.e., ( 11, 9 ). (b) (i) { x : x R, x < 8} is the set that contains but not contain 8.So, it can be represented as an interval whose first end point is closed and the other end point is open i.e., [, 8). On the real li [, 8) may be graphed as shown in figure given below The dark portion on the number li is the required set. (ii) { x : x R, 5 x 6} is the set which contains 5 and 6 both. So, it is equivalent to a closed interval i.e., [5, 6]. On the real li [5, 6] may be graphed as shown in the figure given below The dark portion on the number li is the required set. niversal Set If there are some sets under consideration, then there happens to be a set which is a superset of each o of the given sets. Such a set is known as the universal set and it is denoted by. e.g., (i) Let = {, 4, 6}, = {1, 3, 5} andc = {0, 7} b b 0 8 0 5 6 Then, = {0, 1,, 3, 4, 5, 6, 7} is an universal set. (ii) For the set of all integers, the universal set can be the set of rational numbers or the set of real numbers. a a [ a, b] ( a, b] b b Example 3. What universal sets would you propose for each of the following? (i) The set of right triangles. (ii) The set of isosceles triangles. Solution (i) The universal set for the set of right triangles is set of triangles. (ii) The universal set for the set of isosceles triangle is set of equilateral triangles or set of triangles. Power Set The collection of all subsets of a set, is called the power set of and it is denoted by P(). In P(), every element is a set. e.g., Let = {3, 4, 5} Then, P( ) = {φ, {3}, {4}, {5}, {3, 4}, {3, 5}, {4, 5}, {3, 4, 5}} Note (i) If set has n elements, then the total number of elements in its power set is n. (ii) If is an empty set φ, then P ( ) has just o element i.e., P( ) = { φ }. Properties of Power Sets (i) Each element of a power set is a set. (ii) If, then P( ) P( ). (iii) P( ) P( ) = P( ) (iv) P( ) P( ) P( ) * Method to Write the Power Set of a Given Set Let a set having n elements is given, then for writing its power set, we use the following steps Step I Step II Step III Step IV Write all the possible subsets having single element of given set. Write all the possible subsets having two elements at a time of given set. Write all the possible subsets having three elements at a time of given set. Repeat this process for writing all possible subsets having n elements at a time, as the given set has n elements. Form a set, with the help of the subsets obtaid from steps I, II and III and element φ. This set will give the power set of given set. 10
Sets 11 out Problem If ={ 1,, 3 } then find the power set of. Step I Write all the possible subsets having single element of given set. ll possible subsets of a given set having single element are { 1}, { }, { 3 }. Step II Write all the possible subsets of a given set having two elements at a time. ll possible subsets of a given set having two elements at a time are {1, }, {, 3}, {3, 1}. Step III Write all the possible subsets of a given set having three elements at a time. ll possible subsets of a given set having three elements at a time is {1,, 3}. Step IV Form a set with the help of the subsets obtaid from steps I, II and III and element φ. Hence, the required power set is {{1}, {}, {3}, {1, }, {, 3}, {3, 1}, {1,, 3}, φ}. Example 4. Let = { 1,, 3, 4 }, = { 1,, 3} and C = {, 4 }. Find all sets X satisfying each pair of conditions (i) X and X C (ii) X, X and X C Sol. Given, = { 1,, 3, 4 }, = { 1,, 3} and C = {, 4} Now, P( ) = { φ, { 1 }, { }, { 3 }, { 4 }, { 1, }, { 1, 3 }, { 1, 4 }, {, 3 }, {, 4 }, { 3, 4}, { 1,, 3 }, { 1,, 4 }, { 1, 3, 4 }, {, 3, 4 }, { 1,, 3, 4 }...(i) P( ) = { φ, { 1}, { }, { 3}, { 1, }, { 1, 3 }, {, 3 }, { 1,, 3 }...(ii) and P( C ) = { φ, { }, { 4}, {, 4 }}...(iii) (i) Now, X and X C X P( )and X P( C ) From Eqs. (ii) and (iii), we get X = { 1}, { 3}, { 1, }, { 1, 3}, {, 3 }, { 1,, 3} (ii) Now, X, X and X C X P( ), X, X P( C ) From Eqs. (ii) and (iii), we get X = { 1}, { 3}, { 1, }, { 1, 3}, {, 3} Here, X does not contain = { 1,, 3} as X. EXM Very Short nswer Type Questions [1 Mark each] 1. Consider the following sets φ, = {1, } and = {1, 4, 8} Insert the following symbols or between each of the following pair of sets. (i) φ... (ii)... Sol. (i) Since, null set is proper subset of every set. φ [1/] (ii) Given, = {1, } and = { 1, 4, 8 }. Since, element. [1/]. Prove that φ implies = φ. Sol. Given, φ...(i) ut φ [empty set is a subset of each set] (ii) [1/] From Eqs. (i) and (ii), we get = φ [1/] 3. Write the following subset of R as interval. lso find the length of interval and represent on number li. { x : x R, 1 x 10} If iqualities are of the form or, then use the symbol of closed interval and then find the length of the interval, which is equal to the difference of its extreme values. Sol. { x : x R, 1 x 10} = [ 1, 10] and length of interval = 10 ( 1) = On the real li set [ 1, 10] may be graphed as shown in figure given below The dark portion on the number li is the required set. 4. Let = {a, b, {c, d}, e}. Which of the following statements is/are true? (i) {c, d} (ii) {{c, d }} Sol. Given, = {a, b, {c, d}, e} 1 10 (i) Since a, b, {c, d} and e are elements of. {c, d} Hence, it is a true statement. [1/] (ii) s{ c, d } and{{ c, d }} represents a set, which is a subset of. {{c, d}} Hence, it is a true statement. [1/] 0 11
1 ll Mathematics Class 11th 5. Write down the subsets of the following sets. (i) {1,, 3} (ii) {φ} Sol. (i) The subsets of {1,, 3} are φ, {1}, {}, {3}, {1, }, {, 3}, {1, 3}, {1,, 3}. [1/] (ii) Clearly, { φ} is the power set of empty set φ. Now, its subsets are φ and {φ}. [1/] 6. Let = {1, 3, 5} and = {x : x is an odd natural number < 6}. (i) Is? (ii) Is =? Sol. Given, = {1, 3, 5} and = {x : x is an odd natural number < 6} = {1, 3, 5} (i) Yes, here,because all elements of set are present in set. [1/] (ii) Yes, here,, because all elements of set are present in set. Hence, =, because both sets contain equal and same elements. [1/] 7. Write down the power set of the following sets. (i) = {0, 1, 3} (ii) C = {1, {}} Sol. (i) Given, = {0, 1, 3} P( ) = {φ, {0}, {1}, {3}, {0, 1}, {0, 3}, {1, 3}, {0, 1, 3}} [1/] (ii) Given, C = { 1, { }} P( C ) = {φ, {1}, {{}}, {1, {}}} [1/] 8. In each of the following, determi whether the statement is true or false. If it is true, then prove it. If it is false, then give an example. (i) If x (ii) If and, then x. and C, then C. Sol. (i) False, Let = { }, = {{}, 3} Clearly, and, but. So, x and ed not imply that x. [1/] (ii) False, Let = { }, = {, 3} and C = {{, 3}, 4} Clearly, and C, but C. Thus, and C ed not imply that C. [1/] 9. Given, the sets = {1, 3, 5}, = {, 4, 6} and C = {0,, 4, 6, 8}. Which of the following may be considered as universal set(s) for all three sets, and C? (i) φ (ii) {0, 1,, 3, 4, 5, 6, 7, 8, 9, 10} Sol. We know that, universal set for sets, and C is superset of, andc i.e., universal set contains all elements of, and C. (i) φ cannot be considered as universal set. [1/] (ii) {0, 1,, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the given sets, and C as all the elements of sets, and C are in this set. [1/] Short nswer Type Questions [4 Marks each] 10. If = {3, {4, 5}, 6}, then find which of the following statements are true. (i) { 4, 5} (ii) { 4, 5} (iii) φ (iv) { 3, 6} Sol. Given, = {3, {4, 5}, 6} Here, 3, {4, 5}, 6 all are elements of. (i) { 4, 5}, which is not true. Since, element of any set is not a subset of any set and here { 4, 5} is an element of. (ii) { 4, 5}, is a true statement. (iii) It is always true that φ. (iv) {3, 6} makes a set, so it is a subset of i.e., { 3, 6}. 11. Let and be two sets. Then, prove that = and. Sol. Given = To prove and Proof y definition of equal sets, every element of is in and every element of is in. and Thus, = and gain, let and y the definition of a subset, if then it follows that every element of is in and if then it follows that every element of is in. = Hence, = and Hence proved. 1
Sets 13 1. Exami whether the following statements are true or false. (i) {a, b} {b, c, a} (ii) {a, e} {x : x is a vowel in the English alphabet} (iii) {a} {a, b, c} (iv) {0,1,,3,4,5,6,7,8,9,10} is the universal set for the sets {1, 3, 5} and {, 4, 6}. Sol. (i) Since, the elements of the set {a, b} are also present in the set {b, c, a}. So, { a, b} { b, c, a} {a, b} {b, c, a} is false. (ii) Vowels in the English alphabets are a, e, i, o, u. {a, e} {x : x is a vowel in English alphabet}is true. (iii) False, since a {a, b, c} and not {a}. (iv) True, since all elements of both sets {1, 3, 5} and {, 4, 6} are present in {0, 1,, 3, 4, 5, 6, 7, 8, 9, 10}. Hence, {0, 1,, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for given two sets. 13. Show that n { P [ P ( P ( φ ))]} = 4. Sol. We have P ( φ ) = { φ} P ( P ( φ )) = { φ, { φ }} P [ P ( P φ ))] = { φ, { φ}, {{ φ}}, { φ, { φ }}} Hence, number of elements in P [ P ( P ( φ))] is 4. i.e., n { P [ P ( P ( φ ))]} = 4 Hence proved. TRY YORSELF Directions (Q.Nos. 1 to 3) Determi whether the statement is true or false. 1. If = {3, 6, 7}, = {, 3, 7, 8, 10}, then.. If = {x : x + 4x 1= 0, x N} and = { 7, 3 }, then. 3. If = { 1, 7, 9}, = {, 4, 6} and C = { 0,, 4 }, then = { 0, 1,, 3, 4, 5, 6, 7, 8, 9} is the universal set for all three sets. 4. Let = {4, 5, {6}, 7}, then which of the following statements are incorrect and why? (i) { 6} (ii) { 6} 5. Find the power set of a set = {0, 1, }. 6. If set = {1, 3, 5}, then find the number of elements in P{ P( )}. 7. If = { x : x = n, n = 1,, 3 }, then find the number of proper subsets. 8. Write the following intervals in the set-builder form. (i) ( 6, 0) (ii) [3, 1) (iii) [, 1 ] (iv) ( 3, 5] 9. Write the following as intervals. (i) { x: x R, 3 < x 7 } (ii) { x: x R, 11< x < 7} (iii) { x: x R, 0 x < 11 } (iv) { x: x R, x 9} 13
3 Venn Diagrams and Operations on Sets Venn Diagrams Venn diagrams are named after the English logician, John Venn (1834-1883). Venn diagrams represent most of the relationship between sets. These diagrams consist of rectangles and closed curves usually circles. In Venn diagrams, the universal set is represented by a rectangular region and its subset are represented by circle or a closed geometrical figure inside the universal set. lso, an element of a set is represented by a point within the circle of set. e.g., If = { 1,, 3, 4,..., 10} and = {1,, 3}, then its Venn diagram is as shown in the figure 5 4 6 10 1 Example 1. Draw the Venn diagrams to illustrate the following relationship among sets E, M and, where E is the set of students studying English in a school, M is the set of students studying Mathematics in the same school and is the set of all students in that school. (i) ll the students who study Mathematics also study English, but some students who study English do not study Mathematics. (ii) Not all students study Mathematics, but every student studying English studies Mathematics. Firstly, make a relation between sets under given condition. Then, it is easy to draw a Venn diagram. Solution Given, E = Set of students studying English M = Set of students studying Mathematics = Set of all students (i) Since, all of the students who study Mathematics also study English, but some students who study English do not study Mathematics. M E Through, Venn diagram we represent it as M 3 E 7 9 8 (ii) Since, every student studying English studies Mathematics. Hence, E M Through, Venn diagram we represent it as Operations on Sets There are some operations which when performed on two sets give rise to another set. Here, we will defi certain operations on set and exami their properties. 1. nion of Sets Let and be any two sets. The union of and is the set of all those elements which belong to either in or in or in both. It is denoted by and read as union. The symbol is used to denote the union. = { x : x or x } e.g., Let = {, 3} and = { 3, 4, 5 }, then = {, 3, 4, 5} The union of sets and is represented by the following Venn diagram. The shaded portion represents. Some Properties of nion of Sets (i) = [commutative law] (ii) ( ) C = ( C) [associative law] E (iii) φ = [law of identity element, φ is the identity of ] (iv) = [idempotent law] (v) = [law of ] M C 14
Sets 15 Example. Find the union of each of the following pairs of sets. (i) = { a, e, i, o, u}, = { a, c, d} (ii) = { 1, 3, 5}, = {, 4, 6} (iii) = { x : x is a natural number and 1 < x 5} and = { x : x is a natural number and 5 < x 10} Firstly, convert the given set in roster form, if it is not given in that. Then union of two sets, is the set which consists of all those elements which are either in or in. Solution (i) = { a, e, i, o, u }, = { a, c, d } = { a, c, d, e, i, o, u} (ii) = { 1, 3, 5}, = {, 4, 6} = { 1,, 3, 4, 5, 6} (iii) = { x : x is a natural number and 1< x 5} = {, 3, 4, 5} = { x : x is a natural number and5 < x 10} = { 6, 7, 8, 9, 10} = {, 3, 4, 5} { 6, 7, 8, 9, 10} = {, 3, 4, 5, 6, 7, 8, 9, 10}. Intersection of Sets Let and be any two sets. The intersection of and is the set of all those elements which belong to both and. It is denoted by and read as intersection. The symbol is used to denote the intersection. = { x : x and x } The intersection of sets and is represented by the following Venn diagram. The shaded portion represents. e.g., Let = {, 3, 4, 5} and = { 1, 3, 6, 4} Then, = { 3, 4} Some Properties of Intersection of Sets (i) = [commutative law] (ii) ( ) C = ( C) [associative law] C (iii) φ = φ, = [law of φ and ] (iv) = [idempotent law] (v) If, andc are any three sets, then (a) ( C) = ( ) ( C) [distributive law i.e., distributes over ] This can be shown with the Venn diagram as given below For LHS For RHS C (i) Shaded portion represents ( C) C (iii) Shaded portion represents ( ) (v) Shaded portion represents ( ) ( C) (b) ( C) = ( ) ( C) [ distributes over ] Example 3. (i) If = { 1, 3, 5, 7, 9} and = {, 3, 6, 8, 9 }, then find. (ii) If = { e, f, g} and = φ, then find. (iii) If = { x : x = 3n, n Z} and = { x : x = 4 n, n Z}, then find. Solution (i) Given, = { 1, 3, 5, 7, 9} and = {, 3, 6, 8, 9} = { 3, 9} [Q3 and 9 are only elements which are common] (ii) Given, = { e, f, g } and = φ = φ [since, there is no common element] C (ii) Shaded portion represents ( C) C C (iv) Shaded portion represents ( C) 15
16 ll Mathematics Class 11th (iii) Let x x and x 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 1. x = 1 n, n Z Hence, = { x : x = 1n, n Z } 3. Difference of Sets Let and be any two sets. The difference of sets and in this order is the set of all those elements of which do not belong to. It is denoted by and read as minus.the symbol is used to denote the difference of sets. = { x : x and x } Similarly, = { x : x and x } The difference of two sets and can be represented by the following Venn diagram The shaded portion represents the difference of two sets and. e.g., Let = { 1,, 3, 4, 5} and = { 3, 5, 7, 9} Then, = { 1,, 4} and = { 7, 9} Some Properties of Intersection of Sets (i) = (ii) = (iii) (iv) Example 4. (i) If X = {a, b, c, d, e, f} and Y = { f, b, d, g, h, k}, then find X Y and Y X. (ii) If = { 1,, 3, 4, 5} and = {, 4, 6 }, then find and. Solution (i) Given, X = { a, b, c, d, e, f } and Y = { f, b, d, g, h, k} X Y = { a, b, c, d, e, f } { f, b, d, g, h, k} = { a, c, e} [only those elements of X which do not belong to Y ] and Y X = { f, b, d, g, h, k} { a, b, c, d, e, f } = { g, h, k} [only those elements of Y which do not belong to X ] (ii) Given, = { 1,, 3, 4, 5} and = {, 4, 6} = { 1,, 3, 4, 5} {, 4, 6} = { 1, 3, 5} [only those elements of which do not belong to ] and = {, 4, 6} { 1,, 3, 4, 5 } = { 6} [only those elements of which do not belong to ] 4. Symmetric Difference of Two Sets Let and be any two sets. The symmetric difference of and is the set ( ) ( ). It is denoted by and read as symmetric difference. The symbol is used to denote the symmetric difference. = ( ) ( ) = { x : x or x but x } The symmetric difference of sets and is represented by the following Venn diagram Hence, the shaded portion represents the symmetric difference of sets and. e.g., Let = { 1,, 3, 4} and = { 3, 4, 5, 6} Now, = { 1, } = { 5, 6} = ( ) ( ) = { 1,, 5, 6} * Method of Finding Symmetric Difference of Two Sets If two sets and (say) are given, then we find their symmetric difference with the help of following steps Step I Write the given sets in tabular form (if not given in tabular form) and assume them and (say). Step II Find the difference of sets and i.e.,. Step III Find the difference of sets and i.e.,. Step IV Find the union of sets obtaid from steps II and III, which will give the required symmetric difference. out Problem Find the symmetric difference of sets = { 1, 3, 5, 6, 7 } and = { 3, 7, 8, 9 }. Step I Write the given sets in tabular form Given sets are = { 1, 3, 5, 6, 7} and = { 3, 7, 8, 9} which are in tabular form. Step II Find the difference of sets and i.e.,. We know that, difference of sets and is the set of those elements of, which are not present in. So, = { 1, 3, 5, 6, 7} { 3, 7, 8, 9 } = { 1, 5, 6} 16
Sets 17 Step III Find the difference of sets and i.e.,. We know that, difference of sets and is the set of those elements of, which are not present in. So, = { 3, 7, 8, 9} { 1, 3, 5, 6, 7 } = { 8, 9} Step IV Find the union of sets obtaid from steps II and III, which will give the required symmetric difference. Required symmetric difference, = ( ) ( ) = { 1, 5, 6} { 8, 9 } = { 1, 5, 6, 8, 9} Example 5. If = { 1, 3, 5, 7, 9} and = {, 3, 5, 7, 11 }, then find. Firstly, determi the sets and and then find the union of these sets. Solution We know that, is the symmetric difference of sets and. = ( ) ( ) Given, = { 1, 3, 5, 7, 9}, and = {, 3, 5, 7, 11} Then, = { 1, 9} and = {, 11} = ( ) ( ) = { 1, 9} {, 11 } = { 1,, 9, 11} Disjoint Sets Two sets and are said to be disjoint sets, if they have no common element i.e., = φ. The disjoint of two sets and can be represented by the following Venn diagram Hence, separation of sets represents the two disjoint sets. e.g., Let = {, 4, 6} and = { 1, 3, 5} Then, = φ. Hence, and are disjoint sets. Note The sets, and are mutually disjoint sets i.e., the intersection of any of these two sets is an empty set. Example 6. Which of the following pairs of sets are disjoint? (i) = { 1,, 3, 4, 5, 6} and = { x : x is a natural number and 4 x 6} (ii) = { x : x is the boys of your school} = { x : x is the girls of your school} Firstly, convert all the sets in roster form, if it is not given in that. Then use the condition for disjoint sets i.e., = φ. Solution (i) Given, = { 1,, 3, 4, 5, 6} and = { 4, 5, 6} = { 1,, 3, 4, 5, 6} { 4, 5, 6} = { 4, 5, 6} φ Hence, this pair of sets is not disjoint. (ii) Here, = { b1, b,...} and = { g1, g,... }, where b1, b,..., are the boys and g1, g,, are the girls of school. Clearly, = φ Hence, this pair of set is disjoint set. Complement of a Set Let be the universal set and be any subset of, then complement of with respect to is the set of all those elements of which are not in. It is denoted by or and read as complement. Thus, = { x : x and x } or = The complement of set is represented by the following Venn diagram Hence, the shaded portion represents the complement of set. e.g., Let = { 1,, 3, 4, 5, 6} and = {, 4} Then, = = { 1,, 3, 4, 5, 6} {, 4} = { 1, 3, 5, 6} Note If is a subset of the universal set, then its complement is also a subset of. Some Properties of Complement of Sets (i) ( ) = [law of double complementation] (ii) (a) = (iii) (b) = φ [complement law] (a) φ = (b) = φ [laws of empty set and universal set] 17
18 ll Mathematics Class 11th (iv) (a) ( ) = (b) ( ) = [De-Morgan s law] i.e., the complement of the union of two sets is the intersection of their complements [figure (a)] and the complement of the intersection of two sets is the union of their complements [figure (b)]. ll the operations between two sets can be represented in a single Venn diagram, as given below (Portion shaded by vertical lis in circle ) (a) (Shaded Part) Example 7. Let = {1,, 3, 4, 5, 6, 7, 8, 9}, = {, 4, 6, 8} and = {, 3, 5, 7}. Verify that (i) ( ) = (ii) ( ) = se the formulae, ( ) = ( ), = and = and then simplify it. Solution (i) We have, = { 1,, 3, 4, 5, 6, 7, 8, 9} = {, 4, 6, 8} = {, 3, 5, 7} ( ) (Without shaded part) (Portion shaded by vertical Li in both circles) = {, 4, 6, 8} {, 3, 5, 7} = {, 3, 4, 5, 6, 7, 8} Now,( ) = ( ) = { 1,, 3, 4, 5, 6, 7, 8, 9} {, 3, 4, 5, 6, 7, 8} = { 1, 9 }...(i) Now, = = { 1,, 3, 4, 5, 6, 7, 8, 9} {, 4, 6, 8} = { 1, 3, 5, 7, 9} (b) (Portion shaded by vertical lis in circle ) (Portion shaded by horizontal li between and ) and = = { 1,, 3, 4, 5, 6, 7, 8, 9} {, 3, 5, 7} = { 1, 4, 6, 8, 9} = { 1, 3, 5, 7, 9} { 1, 4, 6, 8, 9} = { 1, 9 }...(ii) From Eqs. (i) and (ii), we get ( ) = Hence verified. (ii) Here, = {, 4, 6, 8} {, 3, 5, 7 } = { } ( ) = ( ) = { 1,, 3, 4, 5, 6, 7, 8, 9} { } ( ) = { 1, 3, 4, 5, 6, 7, 8, 9 } (i) Now, = { 1, 3, 5, 7, 9} { 1, 4, 6, 8, 9} = { 1, 3, 4, 5, 6, 7, 8, 9 } (ii) From Eqs. (i) and (ii), we get ( ) = Hence verified. Example 8. If = {, 4, 6, 8, 10}, = { 1,, 3, 4, 5, 6, 7 }, C = {, 6, 7, 10} and = { 1,, 3, 4, 5, 6, 7, 8, 9, 10 }, then verify that (i) ( C) = C (ii) ( C) = ( ) C (iii) ( C) = ( ) C (iv) ( C) = ( ) ( C) (v) ( C) = ( ) ( C) Solution Given, = {, 4, 6, 8, 10}, = {1,, 3, 4, 5, 6, 7}, C = {, 6, 7, 10} and = { 1,, 3, 4, 5, 6, 7, 8, 9, 10} (i) C = { 1,, 3, 4, 5, 6, 7} {, 6, 7, 10} = { 1,, 3, 4, 5, 6, 7, 10} ( C ) = ( C ) = { 1,, 3, 4, 5, 6, 7, 8, 9, 10} { 1,, 3, 4, 5, 6, 7, 10} = { 8, 9 }...(i) Similarly, = = { 8, 9, 10} and C = C = { 1, 3, 4, 5, 8, 9} C = { 8, 9 }... (ii) From Eqs. (i) and (ii), we get ( C ) = C Hence verified. (ii) Here, = {, 4, 6, 8, 10} Now, C = { 1,, 3, 4, 5, 6, 7, 10} ( C ) = { 1,, 3, 4, 5, 6, 7, 8, 10 }...(iii) Now, = { 1,, 3, 4, 5, 6, 7, 8, 10} ( ) C = { 1,, 3, 4, 5, 6, 7, 8, 10} {, 6, 7, 10} ( ) C = { 1,, 3, 4, 5, 6, 7, 8, 10 }...(iv) From Eqs. (iii) and (iv), we get ( C ) = ( ) C Hence verified. 18
Sets 19 (iii) Here, C = { 1,, 3, 4, 5, 6, 7} {, 6, 7, 10} = {, 6, 7} ( C ) = {, 4, 6, 8, 10} {, 6, 7} = {, 6 }...(v) Now, = {, 4, 6, 8, 10} { 1,, 3, 4, 5, 6, 7} = {, 4, 6} ( ) C = {, 4, 6} {, 6, 7, 10} = {, 6 }...(vi) From Eqs. (v) and (vi), we get ( C ) = ( ) C Hence verified. (iv) Here, C = { 1,, 3, 4, 5, 6, 7} {, 6, 7, 10} = {, 6, 7} ( C ) = {, 4, 6, 8, 10} {, 6, 7} = {, 4, 6, 7, 8, 10 }... (vii) Now, ( ) = {, 4, 6, 8, 10} { 1,, 3, 4, 5, 6, 7} = { 1,, 3, 4, 5, 6, 7, 8, 10} and ( C ) = {, 4, 6, 8, 10} {, 6, 7, 10} = {, 4, 6, 7, 8, 10} ( ) ( C ) = { 1,, 3, 4, 5, 6, 7, 8, 10} {,, 4, 6, 7, 8, 10} = {, 4, 6, 7, 8, 10 }...(viii) From Eqs. (vii) and (viii), we get ( C ) = ( ) ( C ) Hence verified. (v) Now, C = { 1,, 3, 4, 5, 6, 7} {, 6, 7, 10} = { 1,, 3, 4, 5, 6, 7, 10} ( C ) = {, 4, 6, 8, 10} { 1,, 3, 4, 5, 6, 7, 10} ( C ) = {, 4, 6, 10 }...(ix) Now, ( ) = {, 4, 6, 8, 10} { 1,, 3, 4, 5, 6, 7} = {, 4, 6} and ( C ) = {, 4, 6, 8, 10} {, 6, 7, 10} = {, 6, 10} ( ) ( C ) = {, 4, 6} {, 6, 10} = {, 4, 6, 10 }...(x) From Eqs. (ix) and (x), we get ( C ) = ( ) ( C ) Hence verified. EXM Very Short nswer Type Questions [1 Mark each] 1. If = { 3, 5, 7, 9, 11 }, = { 7, 9, 11, 13} and C = { 11, 13, 15 }, then find (i) (ii) C Sol. We have, = { 3, 5, 7, 9, 11} = { 7, 9, 11, 13} and C = { 11, 13, 15} (i) = { 3, 5, 7, 9, 11} { 7, 9, 11, 13} = { 7, 9, 11 } [1/] (ii) C = { 7, 9, 11, 13} { 11, 13, 15} = { 11, 13 } [1/]. Find and when = { 1,, 3, 4, 5, 6} and = {, 4, 6, 8, 10 }. Sol. Now, = { 1,, 3, 4, 5, 6} {, 4, 6, 8, 10 } = { 1, 3, 5} [1/] and = {, 4, 6, 8, 10} { 1,, 3, 4, 5, 6} = { 8, 10 } [1/] 3. State whether each of the following statement is true or false. (i) = {, 4, 6, 8} and = { 1, 3, 5} are disjoint sets. (ii) = { a, e, i, o, u} and = { a, b, c, d} are disjoint sets. Sol. (i) We have, = {, 4, 6, 8} and = { 1, 3, 5} Now, = {, 4, 6, 8} { 1, 3, 5} = φ Therefore, and are disjoint sets. Hence, given statement is true. [1/] (ii) We have, = { a, e, i, o, u} and = { a, b, c, d } Now, = { a} i.e., φ Therefore, and are not disjoint sets. Hence, given statement is false. [1/] 4. Prove that( ) ( C) =. Sol. LHS =( ) ( C ) = { ( ) } ( C ) [by De-Morgan s law] = ( ) ( C ) [Q( ) = ] = (( ) ) (( ) C ) = ( ( )) ( C ) = ( ) ( C ) = ( ) = RHS Hence proved. 19
0 ll Mathematics Class 11th 5. Let S = Set of points inside the square, T = Set of points inside the triangle and C = Set of points inside the circle. If the triangle and circle intersect each other and are contaid in a square. Then, prove that S T C = S, by Venn diagram. Sol. Given, S = Set of points inside the square T = Set of points inside the triangle C = Set of points inside the circle Short nswer Type Questions [4 Marks each] 6. If = { a, b, c, d, e, f, g, h}, then find the complement of the following sets. (i) = { a, b, c} (ii) = { d, e, f, g} (iii) C = { a, c, e, g} (iv) D = { f, g, h} Sol. We have, = { a, b, c, d, e, f, g, h} (i) Complement of is. = = { a, b, c, d, e, f, g, h} { a, b, c} = { d, e, f, g, h } (ii) Complement of is. = = { a, b, c, d, e, f, g, h} { d, e, f, g } = { a, b, c, h } (iii) Complement of C is C. C = C = { a, b, c, d, e, f, g, h} { a, c, e, g } = { b, d, f, h } (iv) Complement of D is D. D = D = { a, b, c, d, e, f, g, h} { f, g, h} = { a, b, c, d, e } 7. Find, if (i) = { 1, 3, 4} and = {, 5, 9, 11 }. (ii) = { 1, 3, 6, 11, 1} and = { 1, 6 }. Sol. (i) We have, = { 1, 3, 4} and = {, 5, 9, 11} Then, ( ) = { 1, 3, 4} {, 5, 9, 11} = { 1, 3, 4} and ( ) = {, 5, 9, 11} { 1, 3, 4} = {, 5, 9, 11 } = ( ) ( ) = { 1, 3, 4} {, 5, 9, 11} = { 1,, 3, 4, 5, 9, 11 } (ii) We have, = { 1, 3, 6, 11, 1} and = { 1, 6} ccording to the given condition, the Venn diagram is given below It is clear from the Venn diagram that, S T C = S Hence proved. Then, ( ) = { 1, 3, 6, 11, 1} { 1, 6} = { 3, 11, 1} and ( ) = { 1, 6} { 1, 3, 6, 11, 1 } = φ = ( ) ( ) = { 3, 11, 1} φ = { 3, 11, 1 } 8. If = {, 3, 4, 5, 6, 7, 8, 9, 10, 11 }, = {, 4, 7 }, = { 3, 5, 7, 9, 11} and C = { 7, 8, 9, 10, 11 }, then compute (i) ( ) ( C) (ii) C (iii) C (iv) ( C) Sol. Given, = {, 3, 4, 5, 6, 7, 8, 9, 10, 11 }, = {, 4, 7 }, = { 3, 5, 7, 9, 11} and C = { 7, 8, 9, 10, 11} (i) ( ) = {, 4, 7} {, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {, 4, 7} C = { 3, 5, 7, 9, 11} { 7, 8, 9, 10, 11} = { 3, 5, 7, 8, 9, 10, 11} ( ) ( C ) = {, 4, 7} { 3, 5, 7, 8, 9, 10, 11} = { 7 } (ii) C = { 7, 8, 9, 10, 11} { 3, 5, 7, 9, 11} = { 8, 10 } (iii) C = { 3, 5, 7, 9, 11} { 7, 8, 9, 10, 11} = { 3, 5 } (iv) ( C ) = ( C ) = {, 3, 4, 5, 6, 7, 8, 9, 10, 11} { 3, 5} = {, 4, 6, 7, 8, 9, 10, 11 } 9. Verify( ) =, where = { 3, 4, 5, 6} and = { 3, 6, 7, 8} are subsets of = { 1,, 3, 4, 5, 6, 7, 8 }. Sol. We have, = { 3, 4, 5, 6 }, = { 3, 6, 7, 8} T and = { 1,, 3, 4, 5, 6, 7, 8} Now, = { 3, 4, 5, 6} { 3, 6, 7, 8} = { 3, 6 } C S 0