1) Express each of the following statements in set theoretical notation:
A) x belongs to the set A.
B) Set A is a subset of set B.
C) P is a proper subset of Q.
D) a is not an element of set X
E) Y is not a proper subset of Z.
F) A Set whose elements are contained in U but not contained in B.
2) Suppose P and Q are two given sets and the corresponding Universal set is U; Express each of the following statement in symbols :
A) A set whose elements are contained either in P or in Q.
B) A set which contains the common elements of P and Q.
C) A set whose elements belongs to U but do not belong to Q.
D) A set whose elements belongs to Q but do not belong to P.
E) A set whose elements do not belong to any of the set P and Q.
F) A set which does not contain the common elements of P and Q.
3) Represent the following sets in Roster form :
A) set of vowels in English alphabet.
B) set a factors of the number 36.
C) A={x: x is a prime integer and 6< x ≤ 29}.
D) P={x| x ∈ N and x≤ 12}.
E) set of letters in the word 'statistics'.
4) Rewrite the following set in set builder notation form:
A) A={......, -3,-2,-1,0,1,2,3}
B) P={1,3,5,7,9,11}
C) set of roots of the equation x⁴ - 13x²+ 36= 0.
D) set of even positive integers greater than 4 and less than or equals to 19
5) State weather each of the following sets is infinite or finite:
A) D={x: x is the number of people living on the earth}
B) W={x:x is the time a person waits for a bus}
C) V={x:x is an odd integer exceeding 889}.
6) State with reasons whether the sets defined in each of the following are equal sets:
A) X= {∅} ; Y= ∅.
B) A={-2, -5}.
B={x:x is a root of the equation x²+ 7x + 10= 0}.
C) P={b, i, e, n, s, u}
Q={x:x is a letter in the word ' Business'.
D) A={x:x is a digit in the number 30255}
B={x:x is an integer and 0≤x≤5}
C={1,0,2,3,4,5}.
7) Some well-defined sets are given below; identify the null sets:
A) A={0}
B) B={∅}
C) C= ∅
D) Set of boy students in a girls' college.
E) P={x:x is an integer and 1<x<2}
F) Q={x:x is an integer and 1<x≤2}
G) R={x:x is positive and x² + 7x +12= 0}
H) A={x:x is an integer and 6x² - 5 +1= 0}
I) {x: 3x² -4= 0, x is an integer}
M) {x: (x+3(x+3)= 9, x is a real number}
N) A∩ B - A.
8) State with reasons which of the following statements are correct/ incorrect:
A) If P U Q={a, b, c, d}, then a ∈ P And a ∈ Q
B) If P∩ Q={a,b,c,d}, then a ∈ P and a ∈ Q.
C) If A∩ B=∅, then A and B are disjoint sets.
D) If A is super subset of B, then B is super subset of A.
E) If A is super subset of B, and B is super subset of C, then A is the super subset of C.
F) x ∈ A U B => x ∈ A.
G) If A is super subset of B, and B is super subset of A, then A = B.
G) If A= {2,4, 6, 8}, then {2,4}∈A.
H) If A= {2,4, 6, 8}, then {2,6,8} is subset of A.
I) If A= {2,4, 6, 8}, then ∅ is the subset of A.
M) If A= {2,4, 6, 8}, then {2,4}is the subset of P(A).
N) (A - B), A ∩ B and (B - A) are mutually disjoint.
O) If A= {2,4,6,8}, then {2,6,8} ∈ P(A).
9) If a ∈ A and a ∈ B, does it follow A is super subset of B ? Give reasons.
10) State with the reasons, which of the following statements are true or false :
A) {a} ∈ {a, b, c}
B) a ∈ {a, b, c}.
C) a⊂ {a, b, c}
D) a not belong to {a, b, c}.
E) 3 ⊂ {1,3,5}
F) 3 ∈ {1,3,5}
G) {3} ⊂ {1,3,5}
H) {3} ∈ {1,3,5}
11) If A = {a,b,c} Name
A) the subsets of A
B) the proper subsets of A.
12) Define power set of a set A. Find the power state of a A{a, b c}, If B be the power set of A, state with the reason, which of the following statement is correct:
A> B, A ∈ B, A⊂B, A= B, A is not subset of B.
13) Fill in the gaps:
A) The number of subsets in a set consisting of four distinct elements are -----.
B) The number of proper subsets in set consisting of n district elements are----.
C) If x ∈ A => x ∈ B then ----.
D) if A is super subset of B and B is super subset of A then-----.
E) the set of P whose elements are all subsets of the set {1,2} is given by
P={__, ___, ___, ___}
F) if A and B are disjoint set then n(A U B)= ____.
G) If A U B= A ∩ B then____.
H) The dual of A U (B ∩C)= (A U B) ∩ (A U C) is _____.
I) The dual of A U U= U is ___.
J) If A is a given set and ∅ is the null set then ___.
14) Let A= {a, b, c}, B={a,b}, C={a,b,d}, D{c, d} and E={d}. State which of the following statements are correct and give reasons:
A) B ⊂ A
B) E is no subset of E
C) D⊂B
D) {a} ⊂A
15) Let A={a,b, c,d,e,f,g,h,i},
B={b,d,f,h}.
C={a,c,e,g,i}, D{c,d,e}, E={c,e}. Which set can equal X if we are given the following informations?
A) X and B are disjoint
B) X ⊂A but X is not subset of C.
C) X ⊂ D but X is not subset of B.
D) X⊂ C but X is not subset of A.
16) List the sets A U B, A ∩ B A∩ (B U C). given that,
A={p,q,r,s}, B={q,r,s,t}, C={q,r,t}
17) If P{a,b,c,d,e} and Q={a,e,i,o,u}, prove P⊂ P U Q and P ∩Q ⊂ P.
18) If A= {2,3,4,5} and B={1,2,3,4} show that B - A ≠ A- B.
19) Let S={1,2,3,4,5} be the Universal set and let A={3,4,5} and B={1,4,5} be two of its subsets. Verify (AU B)'= A' ∩B'.
20) If A{1,2,3,4}, B={2,3,4,5}, C={1,3,4,5,6,7}, find
A) A - B
B) A - C
and hence verify A - (B∩C)= (A - B) U (A - C).
21) If P={a,b, c, d, e, f} and Q={a, c,e,f}, show (P - Q) U (P ∩Q)= P.
22) If A={x : x is an integer and 1 ≤ x ≤ 10} and B={x: x is a multiple of 3 and 5 ≤ x ≤ 30}; find A U B, A∩B, A - B and B - A.
23) Let U={1,2,3,4,5,6,7,8,9,10} be the universal set. suppose, A={1,2,3,4,5,6} and B {5,6,7} are its two subsets. Write down the elements of A - B and A ∩ B'.
24) If X={x : x is an even integer and 6< x ≤ 20} and Y={x : x is a multiple of 3 and 0≤ x≤25}, find
A) X U Y
B) X ∩ Y
C) Y - X
D) X - Y
25) Let S={1,2,3,4,5,6} be the Universal set, let A U B= {2,3,4}; find A' ∩ B' where A' , B' are complements of A and B respectively. also show that A UB, and A' ∩ B' are disjoint sets.
26) if P={p,q,r,s,t,u} and Q={q,r, v,w}; find
A) (P U Q) ∩(P U R)
B) (P - Q)U(P - R).
27) if A,B, C be three subsets of the universal set S where S={1,2,3,4,5,6,7}, A={1,3,5,6} and B ∩C ={1,2,6}, find
A) (A U B)∩(A U C)
B) B' U C'
28) Given, A={x: 0< x≤2} and B={x: 1< x < 3}; find
A) A∩B
B) A U B
C) A - B
29) Let A= {x:2 ≤ x < 5} and B={x : 3 < x < 7} be two subsets of the universal set S={x: 0 < x≤ 10}; verify that (A UB)'= A'∩B'.
30) Given X U Y={1,2,3,4}, X U Z = {2,3,4,5}, X ∩Y={2,3} and X∩Y= {2,4}; find X, Y Z.
31) If aN={ax: x ∈N}, Describe 3N∩ 7N where N is the set of natural numbers.
32) Using set operation show that the numbers 231 and 260 are prime to each other.
33) Applying set operation prove that 3 + 4 = 7.
34) Using Venn diagram or otherwise, solve the following problems:
In a class of 70 students. Each students has taken either English or Hindi or both. 45 students have taken English and 30 students have taken Hindi. How many students have taken both English and Hindi ?
35) A market research group conducted a survey of 1000 consumers reported that 700 consumers liked product A and 480 consumers liked product B. What is the least number that must have liked both products ?
36) In a town of 60% read magazine A, 25% do not read magazine A but read magazine B. Calculate the percentage of those who do not read any magazine. Also find the highest in the lowest possible figures of those who read magazine B.
37) In a statistical investigation of 1003 families of Calcutta, it was found that 63 families had neither a radio nor a T.V., 794 families had a radio and 187 had a television. How many families in that group had both radio and TV ?
38) Three daily newspaper, E, B, H are published in a certain city. 62% of the citizens read E, 59% read B, 41% read H, 40% read both E and B, 28% read both B and H, 24% read both E and H. Find the percentage of citizens who read all the three papers.
39) In a survey of college students, it was found that 40% use their own books, 50% use library books, 30% use borrowed books, 20% use both their own books and library books, 15% use their own books and borrowed books, 10% use library books and borrowed books, and 4% use their own books, library books and borrowed books. Calculate the percentage of students who do not use a book at all.
40) In a city 3 daily news papers X, Y, Z are published; 65% of the citizens read X, 54% read Y, 45% read Z; 38% read X and Y; 32% read Y and Z; 28% read X and Z; 12% do not read any of these three papers If the total number of people in the City be 10000000, find the number of citizens who read all the three newspapers.
41) A company studies the product preferences of 300 consumers. It was found that 226 liked product A, 51 liked product B, 54 laked product C; 21 liked products A and B, 54 liked products A and C, 39 liked products B and C and 9 likwd all the three products. Prove that, the study results are not correct. [Asume that each consumers liked at least one of the three products]
42) The production manager of Sen, Sarkar and Lahiri company examined 100 items produced by the workers and furnished the following report to his boss.
Defect in measurement 50, defect in colouring 30, defect in quality 25, defect in quality and colouring 10, defect in measurement and colouring 8, defect in measurement and quality 20 and 5 are defective in all respect. The manager was penalised for the report. Using appropriate results of set theory, explain the reason for the penal measure.
43) In a survey of 150 students, it was found that 40 students studied Economics, 50 students studies Mathematics, 60 students studied Accountancy and 15 student studied all the three subjects. It was also found that 27 students studied Economics and Accountancy, 35 students studied Accountancy and mathematics. Find the number who studied only economics and the number who studied none of these subjects.
44) Out of thousand students in a college 540 played football, 465 plyed cricket and 370 played volleyball; of the total 325 played football and cricket, 260 played football and volleyball, 235 played cricket and volleyball, 125 played all the three games. How many students
A) did not play any game
B) played only one game.
C) played just two games.
45) A group of consist of a number of students and each Student of the group can speak at least one of the languages Bengali, Hindi and English. 65 can speak Bengali, 54 Hindi and 37 English; 31 can speak both Bengali and Hindi, 17 both Hindi and English, and 18 both Bengali and English. Determine the greatest and least number of students in the group.
46) An investigator interviewed 100 students to determine their preference for the three drinks; Milk(M), coffee (C) and tea(T). He reported the following:
10 students had all the three drinks M, C and R; 20 had M and C; 30 had C and T, 25 had M and T; 12 had M only, 5 had C only and 8 had T only.Find how many did not take any of the three drinks.
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