SET THEORY. 1. Roster or Tabular form In this form the elements of the set are enclosed in curly braces { } after separating them by commas.

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1 SETS: set is a well-defined collection of objects. SET THEORY The objects in a set are called elements or members of the set. If x is an object of set, we write x and is read as x is an element of set or x belongs to set. If y is an object, which does not belong to set, we write y and is read as y is not an element of set or, y does not belong to set. Example: The collection of all natural numbers greater than 500 is a set. The collection of vowels in English alphabets is a set. ut The collection of all difficult problems in mathematics is not a set. (why?) The first two are sets because the properties are well defined here and we are clear if any object is there whether that will belong to this collection or not but in the last example the collection of difficult problems in math s is not a set because the same problem may be hard or simple for two different person. Notations There are mainly two ways for representing a set. 1. Roster or Tabular form In this form the elements of the set are enclosed in curly braces { } after separating them by commas. Example: (i) the set of integers (Z) = {., -, -, -1, 0, 1,,,.} (ii) the set of whole numbers (W) = { 0, 1,,, } (iii) the set of natural numbers (N) = { 1,,,, } (iv) the set of odd natural numbers = {1,, 5, 7, } (v) the set of prime numbers less than 8= {,, 5, 7} Note: 1. The change of order of writing the elements of a set does not change the set. i.e. {1,,} = {,,1} = {, 1, }.. The elements of a set are never repeated; i.e. an element in the set should be written only once. e.g. {1, 5, 1,, 5, 1} = {1,, 5}.. Set-uilder form In this form, a rule or a statement in the briefest possible way is given. With the help of this rule or statement, the elements of the set can be found. Example: Let be the set of capital cities of India, then set is not written by actually writing the names of the capital cities of India, but we write: = { x : x is a capital city of India } or {x/x is a capital city of India} This is read as is the set of x such that x is a capital city of India. Each of the symbols : and / is read as such that.

2 Cardinal Number of a Set The number of elements in a set is called its cardinal number. Example: Set = {1,,, } has elements, so its cardinal number is and we write n() =. KINDS OF SETS Finite Set set with finite number of elements in it is called a finite set. Example: (i) Set of mountains in India. (ii) {,, 6, 8, 10,.80} i.e set of positive even not less than 80 etc. Infinite Set set which is not finite, is called an infinite set. Example: (i) Set of points in a line (ii) {1,,,..} (iii) {x : x Z and x > 5 } etc. Singleton or Unit set set, which contains only one element in it, is called a singleton or unit set. Example (i) Set of whole numbers between and 5. We know that there is only one whole no between and 5 i.e so our answer is {} similarly (ii) { x : x + = 5} etc. Empty Set or Null Set The set, with no element in it, is called the empty set and is denoted by { } or Empty set is unique set. Example: = { x : x N and < x < 5 } as we know that there is no natural no between and 5 so the set will be empty or having no element Note: 1. The symbol denotes the empty set.. The sets, {0} and {} are all different sets. Since, = the empty set with no element in it; {0} = singleton set with 0 as its element and {} = the singleton set with the empty set as its element.. n() = 0 SET RELTIONS Joint or Overlapping Sets Two sets are said to be joint or overlapping sets if they have atleast one element in common. Example: Set = {1,,, } and et = {,, 5, 6} are joint sets, since the elements and are common in both the sets. Disjoint Sets Two sets are said to be disjoint, if they have no element in common.

3 Example: Set = {1,, 5} and set = {,, 6} are disjoint, since they have no element in common. Equivalent Sets Two sets are equivalent, if they have equal number of elements, i.e. if the cardinal numbers of two sets are equal, the sets are equivalent. Example: Set = {a, b, c, d, e} and set = {1,, 9, 16, 5} are equivalent sets as n() = n() = 5. When set is equivalent to set, we write: Two infinite sets are always equivalent sets Equal Sets Two sets are equal, if they have same (identical) elements. Example: (i) If = {1, 5, 7} and = {1, 1, 5, 7}, then =. (ii) If P = {1, } and Q = {x : x 5x + = 0}; then P = Q. Equal sets are always equivalent, but equivalent sets are not necessarily equal. Exercise:1 (i) Which of the following are sets: (a) Some even numbers (b) Collection of Integers (ii) Write the following sets in Roster Form (a) = { x : x is a two digit natural number such that sum of its digits is 7} (b) = The set of all letters of word FIITJEE (iii) Write the set builder form: (a) {, 6, 9, 1} (b) {6, 6, 16, 66} (iv) Find whether = or not? (a) = {,, 6, 8, 10} = {x : x is positive even integer less than or equal to 10} (b) = {x : x is a multiple of 10} = {10, 15, 0, 5, 0} (v) Find whether the set is equivalent to set. (a) = set of all even integers = set of all odd integers. (b) = { x : x is a natural no. divisor of 6} = { x : x is a divisor of 8}. Subset Set is said to be a subset of set, if every element of set is also an element of set. we write it as and read as is subset of or is contained in. Example: (i) If = {, 5, 6} and = {1,,,,.., 10} then; is subset of i.e.. (ii) If = set of girls under 6 years and = set of girls under 0 years, then is Subset of i.e.. Note: 1. Every set is a subset of itself i.e.,, C C and so on.. Empty set is subset of every set.. If a set has n elements in it, the number of its subsets = n.. If and, then =.

4 Super Set If a set is a subset of set then is called super set of. It is written as and is read as is super set of Proper Subset set is said to be a proper subset of set, if every element of is in and has at least one extra element than set. Example: If = {1,, } and = {1,,, }, then is proper subset of. It is written as and is read as is proper subset of. Note: 1. No set is proper subset of itself.. Empty set is proper subset of every set except itself.. If a set has n number of elements in it, the number of its proper subsets = n 1. Power Set Power set of a set is the set of all its subsets. It is denoted by P(). Example If = {a, b}, then its power set P() = {, {a}, {b}, {a, b} }. Universal Set It is the set which contains all elements of the sets under consideration i.e. all the sets under consideration are subsets of he universal set. Universal set is denoted by or (pxi). Example: Let = {1,,, }, = {,, 7, 8} nd C = {,, 6, 8, 10}, then for these sets we can take {1,,., 10 } as their universal set. For the given sets, the choice of universal set is not unique i.e. there can be more universal sets for the same sets under consideration. Example: The set of natural numbers N or the set of whole numbers W or set of integers Z or { x : x N, N 15 } etc. can also be taken as universal set for the sets, and C given above (since each of these sets contains every element of the sets under consideration). In the similar manner more universal sets can be obtained. Complement Set The complement of a set a is the set of elements which belong to the universal set and do not belong to set. It is denoted by. = {X : X and x }. Example: Let universal set = {a, b, c, d, e, f} and set = {a, c, d, f}, then compliment of set = = {b, e}. Note: 1. and are disjoint sets, i.e. a set and its compliment are always disjoint.. = i.e. the complement of the empty set is the universal set.. = i.e. the compliment of the universal set is the empty set.

5 SET OPERTIONS Intersection of Two Sets: The intersection of two sets and is the elements common to both and. It is written as and is read as intersection or cap. = { x : x and x }. Example: Note: Let = { 1,, } and = {,, } the = {, }, since the elements and are common to both and. 1. and.. If =, hen and are disjoint sets.. If, then and are joint i.e. overlapping sets.. Intersection of a set and its complement is always an empty set i.e. =. 5. If, then = and if, then =. 6. If, then. Union of Two Sets: Union of two sets a and is the set of elements which belong to or to or to both. It is written as and is read as union or cup. = {x : x or x or x }. Example: Let = {1,, } and = {,, } then = {1,,, }. 1. and.. If, then = and if, then =.. U = Difference of Two Sets: If and are two sets, then their differences is or, where = Set of elements of, which do not belong to. = {x : x and x }. nd = Set of elements of, which do not belong to = {x : x and x }. Example: Let = {1,,, } and = {,, 5, 6}, then = {1, } and = {5, 6}. 1. ( ) and ( ).. = = -.. =.. ; unless =. 5. = = 6., and are all disjoint sets. Laws of Set Operations: 1. Commutative Laws: For any two sets and : (i) the union of sets is commutative i.e. =. (ii) the intersection of sets is commutative i.e. =.. ssociative Laws: For any three sets, and C: (i) the union of sets is associative i.e. ( ) C = ( C ). (ii) the intersection of sets is associative i.e. ( ) C = ( C ).

6 . Distributive Laws: For any three sets, and C: (i) The union is distributive over intersection. i.e. ( C ) = ( ) ( C) (ii) The intersection is distributive over union, i.e ( C ) = ( ) ( C). De-Morgan s Laws: For any two sets and : (i) The complement of their union is equal to intersection of their complements i.e. ( ) =. (ii) The complement of their intersection is equal to union of their complements i.e. ( ) =. lso remember that: 1. for any two sets and, n ( ) = n() + n() n( ) If and are disjoint sets, then n ( ) = n() + n() [ since = ; n ( ) = 0]. For any three sets, and C n ( C ) =n() + n() + n (C) n( ) n( C) n(c ) + n( C). ( C) = ( ) ( C) and ( C) = ( ) ( C). Exercise : 1. Given P is the set of letters in he word MOON. Find the power set of P.. If = {a, b, c, d}, find the total number of non empty subsets of. Given, the universal set = {.,., 1, 5, 0.5, 1., 7} and = { x : 5 x < } ; = { x : x < 1 } Find and Write all the proper subsets of the set { x : x x = 0} 5 Given universal set = { x W: x 10 }, = {0, 1,, 5,7 } and = { y : y N and y 9}, Then find: (i) (ii) ( ) (iii) ( ) (iv) (v) (vi). Hence show that ( ) = and ( ) =. VENN-DIGRM John Venn, an English Mathematician, has developed the idea of using geometrical figures to represent sets. These figures are called Venn-Diagrams. Euler, a mathematician, also used diagrams to represent sets and so, sometimes, these diagrams are also called Venn-Euler diagrams. In a Venn diagram, generally, the universal set is represented by a rectangle and all other sets under consideration by bounded areas generally a circle within the rectangle. 1. The adjacent diagram shows a universal set,

7 . The adjacent diagram shows and are disjoint, i.e. =. The following figures shows that the sets and are overlapping sets. (i) (ii) (iii) The shaded portion represents.. In the following figures the shaded portion represents. (i) (ii) (iii) 5. Similarly, consider the shaded portion of each of the following venn-diagrams: (i) (ii) ( ) ( ) 6. (i) (ii)

8 7. (i) (ii) ( ) ( ) 8. (i) (ii) ( - ) ( - ) 9. The following diagrams represent subsets, i.e. (i) (ii) ( ) ( ) Illustration 1: - Use Venn-diagrams to prove that ( ) =. Solution: ( ) = (The region which is in or in or in both). ( ) = (The region in the universal set and not in ). I

9 Now, = (The region of universal set, which is not in ). = (The region of universal set, which is not in ). = (The region which is common to both and ). II Since the shaded portions in diagrams I and II represent the same region, ( ) =. (Hence Proved). Illustration : Given universal set = { a, b, c, d, e, f }, = {b, c, d, e} and = {a, b, e,}. Draw Venn-diagram to represent the relationship between the given sets. Use the diagram drawn to find the following sets : (i) (ii) (iii). Solution: The given sets and are overlapping sets, as = {b, e}. Therefore, the Venn-diagram representing the relationship between the given sets will be as shown alongside: (i) = {elements of universal set, which are not in } = {a, f}. (ii) = {elements of, which are not in } = {d, c}. (iii) = {x : x and x } = { a }. f d c b e a Illustration : In X standard of a certain school, students got distinction in Hindi, 50 got distinction in Maths and 7 students in both the subjects. Use Venn diagram to find the number of students getting distinction (i) in Hindi only; (ii) in Maths only; (iii) in Hindi or in Maths or in both ( i.e. total number of distinctions in both the subjects ).

10 Solution: In the Venn diagram drawn alongside; H represents the number of distinctions in Hindi, and M represents the number of distinction in Maths. Since, the portion common to both the circles represents the number of distinctions in both the subjects, therefore, the number 7 is written in this common portion. H() 15 7 M(50) (i) (ii) (iii) The region shaded by horizontal lines represents the number of distinctions in Hindi only. 7 = 15 students got distinction in Hindi only The region shaded by vertical lines represents the number of distinctions in Maths only, 50 7 = students got distinction in Maths only. It is clear from the figure that = 65 students got distinction either in Hindi or in Maths or in both. Exercise : 1. If n() = 50, n() = 7, n() = 19 and n( ) = 11 ; use venn-diagram to find: (i) n( ) (ii) n( ). If, and C are three subsets of the universal set U, draw a venn diagram showing (i) [( ) C] (ii) ( ) C Illustration : Solution: Out of 100 students who appeared for the SSC examination from a school 15 students in English; 1 students in Mathematics; 8 students in Science; 7 students in Mathematics and science; students in English and Science; 6 students in English and Mathematics; students in all the three subjects could get First class marks. How many of them have got First class mark. (i) only in Mathematics (ii) only in science (iii) only in English (iv) exactly in two subjects (v) in more than one subject? The number of students who have got First class Marks (i) Only in Mathematics = n(m) n(e M) n(m S) + n(e M S) = =. (ii) Only in English = n(e) n(e M) n(e S) +n(e M S) = = 9 (iii) Only in science = n(s) (M S) n(s E) + n(m S E) = = 1 (iv) Exactly in two subjects= = 5 (v) In more than one subject =++0+ = 9 n(e)= C 1 n()=100 n(m)=1 n(s)=8 Cartesian Product of Sets: Suppose, are two sets. The sets of all possible ordered pairs (a, b) with a, b is called the Cartesian product of and and is denoted by X (read as cross ). i.e, X = {(a, b) / a, b } We observe the following from definition.

11 = {, } (i) In any ordered pair of X, the first coordinate belongs to while the second belongs to. (ii) X X (iii) If n() = p, n() = q the n(x) = pq. Here n() stands for the number of elements in. (iv) If is empty or is empty, then X is empty. (v) If one of the sets and is infinite the other is non-empty, then X is infinite. Cartesian Product of Two Sets: Suppose, are two sets, then the elements of X can be described in three ways. (i) rrow Diagrams (ii) Tree Diagrams (iii) Graphical representation We shall explain these methods through the following example. Illustration 5: If = {1,, 5}, = {, } then describe X using (i) rrow Diagram (ii) Tree Diagram (iii) Graphical representation. lso observe that X X. Solution: Given = {1,, 5} = {, } X = {1,, 5} X {, } = {(1, ), (1, ), (, ), (, ), (5, ), (5, )} X = {, } X {1,, 5} = {(, 1), (, ), (, 5), (, 1), (, ), (, 5)} It is clear that X X (i) rrow diagram of X 1, 1,, 1, 5, 5 5 (ii) Tree Diagram of X Set Set Elemens of X (1, ) 1 (1, ) (, ) (, ) 5 (5, ) (iii) Graphical representation of 5 1 (5, ) 1 5 = { 1,, 5}

12 = { 1,, 5} (iv) Graphical representation of Dots shows the elements of sets X, X respectively. These representations of X and X illustrate that X X = {, } RELTION Relation is a set of ordered pairs. Every set of ordered pairs is a relation. We use the letter R to denote a relation. Domain and Range of a relation: The set of first coordinates of all the ordered pairs of a relation R is called the domain of R, while the set of second coordinates of all the ordered pairs of R is called the range of R. Illustration 6: If R = {(1, ), (, ), (, ), (, 5)} is a relation, find its domain and range. Solution: Domain of R = the set of all first coordinates of members of R = {1,,, } Range of R = the set of all second coordinates of members of R = {,,, 5} Relation another definition: If, are two sets, we know that X = {(a, b) / a, b } Since ever relation is a set of ordered pairs, a relation can be defined to be a subset of some Cartesian product. If, are two sets, then any subset of R X is said to be a relation from the set into the set. R is a relation from into if and only if R X relation from into is called a relation in. That is if R X, then R is called a relation in Inverse Relation: If R is a relation from a set into another set, then by interchanging the first and the second coordinates of ordered pairs of R we get a new relation from into. This relation is called the inverse relation of R and is denoted by R 1. i.e, (x, y) R if and only if (y, x) R 1. R 1 = {(y, x) / (x, y) R} Note: 1. Domain and Range of R -1 are respectively range and domain of R.. If R is the relation defined by is less than in the set of real numbers, then R 1 is the relation is greater than. Illustration 7: If R = {(, ), (, ), (, ), (, ), (, ), (, )} Is a relation in = {,, }. Find R 1. Solution: R 1 = {(, ), (, ), (, ), (, ), (, ), (, )} Observe that R = R 1.

13 Illustration 8: If R is the relation defined by is factor of in the set = {,, 6} find R 1 Solution: R = {(, ), (, ), (, 6), (, ), (6, 6)} Since is a factor of, and 6 (, ), (, ), (, 6) R nd, 6 are respectively the factors of and 6 only. (, ), (, ), (, 6), (, ), (6, 6) are the elements of R. R -1 = { (, ), (, ), (6, ), (, ), (6, 6)} Types of Relations: 1. One-One Relation: relation r: is said to be one-one relation if no two elements of have the same image in. Example:. One to Many Relation: relation r : is said to be one-to-many relation if an element of is related to two or more elements of. Example: r. Many-One Relation: relation r : is said to be many one relation if two or more elements of are related to an element of. Example: r 1 1 r a. Many- Many Relation: relation r : is said to be many-many relation if two or more elements of are related to two or more elements of. Example: 5 1 r 8 b Types of relations: Suppose R is a relation in a set.i.e. is R X. We shall now discuss certain types of relations in a set. (i) Reflexive relation: R is a relation in and for every a, (a, a) R then R is said to be a reflexive relation. Examples: 1. Every real number is equal to itself. Therefore is equal to is a reflexive relation in the set of real numbers.. If is a set of sets and R is a relation defined by is subset of in, then R is Reflexive as every set is a subset of itself. 6

14 (ii) Symmetric Relation: R is a relation in and (a, b) R implies (b, a) R then R is said to be a symmetric relation. Examples: 1. In the set of all real numbers is equal to relation is symmetric.. is the set of lines in a plane. R is the relation in defined by is parallel. Then R is Symmetric. For if l, m are two lines in (l, m) R then we have l m. ut this implies that m l. (m, l) R. Thus R is symmetric. (iii) nti symmetric relation: R is a relation in. If (a, b) R and (b, a) R implies a = b, then R is said to be an anti-symmetric relation. Examples: 1. In set of all natural numbers the relation R defined by x divides y if and only if (x, y) R is anti-symmetric. For if x y and y x then x = y.. is a set of sets. R is a relation in defined by (X, Y) R if and only if X Y. Then R is anti symmetric. (X, Y), (Y, X) R implies that X Y and Y X, so that X = Y. R is anti-symmetric. (iv) Transitive relation: R is a relation in. If (a, b) R and (b, c) R implies (a, c) R, then R is called a transitive relation. Examples: 1. In the set of all real numbers the relation is equal to is a transitive relation. For a=b, b = c implies a = c.. is the set of all lines in a plane. R is the relation is perpendicular to in. Then R is not a transitive relation. For l m, m n do not imply l n. In fact l n. Thus R is not transitive. (v) Equivalence Relation: relation R in a set is said to be an equivalent relation if it is reflexive, symmetric and transitive. Examples: 1. T is the set of all triangles in a plane. For x, y T, the relation R is defined by x is congruent to y. Then R is an equivalence relation. For x, y T. (a) x x for all x T (b) x y, y z imply x z. (c) x y implies y x.. In the set of all real numbers the relation is equal to is an equivalence relation. For a R, a = a, a = b implies b = a and a = b, b = c implies a = c. Relations and Functions:, are two sets. f is the relation from into. If f is such that for every a there is a unique b such that (a, b) f, then f is said to be a function from into. That is every function is a particular type of relation only. f is a function from into we mean (i) f X (ii) for every a here is a unique b such that (a, b) f.

15 If f is a function from into then we write f : or f and we say that f maps into or f transforms into. function is also called a mapping. If f : is a mapping and (a, b) f, then we write f(a) = b. f(a) is called the image of a. is called the domain of f and is called the co-domain of f. The set f() of all images of elements of under the mapping f is called the range of f. Note: f(). We shall study these functions in detail in class X. We will close the section with some examples. R Example: 1 a R = { (1, a), (, b), (, b), (, c) } Observe that every element of is associated b with exactly one element of. R precisely c contains such ordered pairs only. R is a function from into. Exercise: 1. If = {,, }, = {,, 5} and C = {5, 6} then verify (i ) ( ) x C = ( x C) ( x C) (ii) ( ) x C = ( x C) ( x C). If = {1, }, = {1,, } and C = {,, } then verify ( x C) ( x C). Write the domain and ranges of the following relations. lso find their inverse relations. (i) R 1 = {(1, 1), (, 1 ), (, 1 ), (, 1 )} (ii) R = {(1, 1), (, ), (9, ), (16, )}. Is the relation R = {(1, 1), (, ), (, ) in = {1,, } an equivalence relations? NSWERS TO EXERCISES EXERCISE 1 (i) (a) No (b) Yes (ii) (a) {16, 5,,, 5, 61, 70} (b) {F, I, T, J, E} (iii) (a) {x : x = N, N } (b) {x : x = 5 N + 1, N } (iv) (a) Yes (b) No (v) (a) Yes (b) Yes EXERCISE 1. {, {M}, {O}, {N}, {M, O}, {M, N}, {O, N}, {M, O, N}}. 15. {1., 7}, {., 1.}., { 1}, {} 5. (i) {0, 1} (ii) {10} (iii) {0, 1,,, 6, 8, 9, 10} (iv) {10} (v) {0, 1,,, 6, 8, 9, 10} (vi) {0, 1} EXERCISE 1. (i) 7 (ii) 0 EXERCISE (i) Domain {1,,, }, Range 1,,,, R = (1,1),,,,,, (ii) Domain {1,, 9, 16}, Range {1,,, }, R 1 = {(1, 1), (, ), (, 9), (, 16)}. not an equivalence relation

16 SSIGNMENTS SUJECTIVE LEVEL I 1. State, which of the following collections are sets: (i) Some odd numbers divisible by. (ii) The languages spoken in India. (iii) ll interesting books. (iv) The collection of interesting dramas written by Shakespeare. Write the following sets in set builder form: (i) = {0,, 6, 1, 0 } (ii) = {,, 5, 7, 11} (iii) E is the set whose elements are obtained by adding to each of the odd numbers.. List the elements of the following sets : (i) {x : x N and x x 5 = 0} (ii) {x : x Z and x 16} (iii) {x : x N and x is a factor of } (iv) {x : x = y, y N and y 5}. State, which of the following sets are empty: (i) = {x : x N and x < } (ii) = {x : x + = } 5. State whether the following pairs of sets are equal or not: (i) {5, 0, 5, 0} and {5 x 5, 5 x 6, 5 x 7, 5 x 8} (ii) The set of prime numbers between 10 and 0; and {11, 1, 15, 10} (iii) = {0, 1,,, } and = {x : x N and N } (iv) = {x : x x = 0} and = {x N} 6 State whether the following pairs of sets are equivalent or not: (i) {, 6, 9, 1, 15} and {x : x W and W } (ii) Set of odd natural numbers and set of even natural numbers. (iii) Set of integers and set of multiples of 7. (iv) Set of pupils in a class and set of chairs in the same class. 7. nswer whether the following statements are true of false. Give reasons. (i) If two sets are equal, they are equivalent also. (ii) If two sets are equivalent, they are equal also. (iii) Empty set and set {0} are equivalent. (iv) If is an infinite set, then all its subsets are also infinite. (v) The number of subsets of {x : x + = } is. (vi) The set of even prime numbers is an empty set. (vii) If set has 5 elements, then the number of its proper subsets = 1.

17 (viii) The empty set has no subset. (ix) The total number of subsets of a finite set which contains n elements is n. (x) If and C, then C. 8. If and are two sets such that has 18 elements, has 8 elements, and has 15 elements, how many elements does have? 9. If n() = 5, n() = 6, n() =16. Find the least and greatest possible value of n ( ). 10. If M is the set of letters in the world MHN, find: (i) n(m) (ii) all proper subsets of M. (iii) power set of M LEVEL II 1. In a group of 50 people, 5 speak Hindi, 5 speak both English and Hindi, and all the people speak at least one of the two languages. How many people speak (i) English only, (ii) English. In a group of 70 people, 7 like coffee, 5 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?. In a group of 65 people, 0 like cricket, 10 like both cricket and tennis, How Many like tennis only and not cricket? How many like tennis?. n elocution competition was held in Telugu and Hindi. Out of 80 students 5 students took part in Telugu, 5 in Hindi, 15 both in Telugu and Hindi. Then find the number of students. (i) who took part in Telugu but not hindi (ii) who took part in Hindi but not Telugu (iii) who took part in either Telugu or Hindi (iv) who took part in neither 5. Out of 70 students, 5 play asket-ball; 0 play Football; play Kabadi; 10 play asketball and Football; 9 play asket-ball and Kabadi; 1 play Kabadi and Football and 5 play all the three games. Using Venn diagram find the number of students who play none of these?

18 OJECTIVE LEVEL I 1. If = {x : x 6} and = {x : x 6}, then. () x 6 () x 6 (C) x = 6 (D). ( ) is equal to () ( ) () (C) (D) none of these. = {,, 6, 8}, = {x : x N, N > 9}, then = () () {,, 6} (C) (D) none of these. Given n () = 0, n () = and n () = 0. Find the greatest possible value of n ( ) () 0 () (C) 0 (D) 6 5. If n() =, n() = 6, n() =18. Find the least possible value of n( ). () () 18 (C) 6 (D) None of these 6. If n() = 100, n() = 50, n() =. Find the greatest value of n( ). () 50 () (C) 8 (D) 0 7. If n() = 10, n() = 55, n() =. Find the least value of n ( ). () 55 () (C) 0 (D) None of these 8. ( ) is equal to () () (C) (D) None of these 9. If U = {x : 1 < x < 0, x N}, P = {p : p = n, n N}, Q = {q : q = n, n N}, R = {r : r = 9n, n N}, then P Q R = () {18} () {6, 8} (C) 18 (D) None of these 10. True or False. (i) = (ii) )If P and Q are two overlapping sets then n(p) + n(q) < n (P Q) (iii) If then = (iv) If = then = U (v) {} {,, } (vi) ( ) = (vii) ( ) = ( ) (viii) For any two sets and n( ) = n() + n() n( ) (ix) If then, =

19 1. If every i =, then 10 i1 i LEVEL II () 5 () 10 (C) (D) 100. If x = {g, 0, a, l, s}, then now how many subsets: contain g and s? () 16 () 8 (C) (D). If R is a relation on a set such that R = R 1, then R is () Symmetric () Reflexive (C) ntisymmetric (D) Partial order. Let R be a relation on the set of real numbers defined as a R b iff a + b = 7 then R is () Reflexive () Symmetric (C) Transitive (D) nti Symmetric 5. I: Every relation is a function. II: Every function is a relation. () Only I is true () Only II is true (C) oth I and II are true (D) neither I not II true. 6. = {a, b, c}, {x, y}, f = {(a, x), (b, y), (c, x)}, then f is... mapping. () onto () one-one (C) one-one-onto (D) none of these 7. If c = {0, {1}}, the P(c) = () {, {1}, {0, 1}, {0, {1}}} () {{1}, {0, 1}, {0, {1}}} (C) {, {0}, {{1}}, {0, {1}}} (D) none of these 8, Which of the following is true () a {a, b, c} () {} {a, b, c} (C) {a} {a, b, c} (D) none of these 9, In a class of 60 pupils, 8 play hockey, play cricket and 1 play name of there games. How many play hockey only. () 1 () 18 (C) 1 (D) Null set has () proper sub set () no proper sub set (C) only one element (D) none of these

20 NSWERS SUJECTIVE LEVEL I 1. (i) No (ii) Yes (iii) No (iv) Yes. (i) {x : x = n + n, n w} (ii) {x : x is prime number less than 11} (iii) {x : x = (n +1) +, n w}. (i) {5} (ii) {,,, 1, 0, 1,,, } (iii) {1,,,, 6, 8, 1} (iv) {6, 9, 1, 15} (v) {1,,, }. (i) Yes (ii) No 5. (i) Yes (ii) No (iii) No (iv) No 6. (i) Yes (ii) Yes (iii) No (iv) No 7. (i) T (ii) F (iii) F (iv) F (v) T (vi) F (vii) T (viii) T (ix)t (x) F Least possible value of n( ) = 0, Greatest possible value of n( ) = (i) (ii) Proper subsets = n(m) 1 = 1 = 15, this are, {M}, {}, {H}, {N}, {M,, H}, {M,, N}, {M, H, N}, {, H, N}, {M, }, {M, H}, {M, N}, {, H}, {, H}, {H, N} (iii) P(M) = {, {M}, {}, {H}, {N}, {M,, H}, {M,, N}, {M, H, N},{,H,N}, {M, }, {M, H}, {M, N}, {, H}, {, H}, {H, N},{M,,H,N}} LEVEL II 1. (i) 15 (ii) (i) 5 (ii) 5. (i) 0 (ii) 0 (iii) 60 (iv) OJECTIVE LEVEL I 1. D.. C. 5. C C (i) F (ii) F (iii) T (iv) T (v) F (vi) T (vii) F (viii) T (ix) T (x) T LEVEL II 1. C C 8. C