LAST TIME REVISION SET THEORY (1)

1) Write down the following statements in set-theoretical notations 
a) 3 is an element of a set A.         3 ∈ A
b) 4 does not belong to a set B.       4∉ B
c) C is a set of D.         C⊆ D
d) P and Q are disjoints sets.     P∩ Q = ∅

2)  Represent the following sets in tabular (or Roster) form:
 a) set a factors of 30.      A={1,2,3,4,5,6,10,15,30} 
b) X={a: a is a perfect square and 2 < a≤ 49}.       X={4,9,16,25,36,49} 
c) Y={x: x is an even natural number greater than 20}.        Y={22,24,26,28,30,...}

3) Write the following sets in the set-builder form:
a) set of letters in the word 'statistics'. {x:x is a letter in the word'statistics'} 
b) A= {3,6,9, 12,15,...}.         A={x:x ∈ N and x is a multiple of 3} 
c) Set of integers either equal or greater than 3 but less than 25.       P={x:x is an integer and 3≤ x < 25}

4) Which of the following sets is the null set ∅ ? 
a) A ={x:x is > 1 and x is <1}.             Yes
b) B={x:x+3=3}.                No
c) C={∅} .                               No

5) Show the relationships among the following three sets in respect of subsets and supersets :
a) N={x:x is a positive integer}.           Z⊃N
b) Z={x:x is an integer}.            R⊃N 
c) R={x:x is a real number}.           R⊃Z

6) State whether the sets defined in each of the following cases are equal.
a) x= ∅, Y{∅}.                 Not
b) A={x:x² -3x+2=0} and B={x:x is a digits in the number 212}.     Y
c) P={x: x is an integer and -2≤ x≤2} and Q={x:x(x²-1)(x²-4)=0}.       Y

7) State whether each of the following sets is finit or infinite:
a) A = {x:x is an odd integers greater than 100}.         Inf
b) B={x:x is a real and -1≤ x < 1}.          Inf
c) C={x:x is an odd in negative integer greater than -150)}.    F

8) If A={1,2,3}, then there exist 8 subsets of A. Determine all these subsets of A.          ∅, {1},{2},{3},{1,2},{2,3},{3,1}, A.

9) Given A= {x,y,z}; state which of the following statements are correct:
i) {x} ∈ A.           Inc
b) x ∈ A.        C
c) {x}  ⊂ A.    C
d) y ⊆ A.     Inc
e)  ∅∈ A.         Inc
f) ∅⊆ A.         C
g) {x,y,z} ⊆ A.      C
h) {z} ∈P(A).       C

10) If A={1,2,3,4}, B={2,4,5,6} and C={1,3,4,6, 8}, then find the set A∩ (B ∪C).       

11) If A={1,2,3,4,5,6 ,7,9 ,11,13,15} and B={2,4,6,8, 10, 12, 14 ,16},  find A - B and B- A

12) If A={1,2,3,4}, B={3,4,5} and C={1,4,5}, then verify A- (B∪C)= (A - B)∩(A - C).

13) Given A={1,2,3}, B={2,4}, C={2,3,5}, Find
a) A ∩ B, A∩C and (A∩B)∪(A ∩C)
b) B∪ C and A∩(B∪C),
Hence, verify the result A∩(B∪C)= (A∩B)∪(A∩C).

14) Let the sets A and B be given by, A={1,2,3,4}, B= {2,4,6,8,10} and the universal set S={1, 2 ,3, 4, 5, 6, 7, 8, 9, 10}. Find (A∪B)' and (A∩B)'.

15) If S={a,b,c,d,e,f} be the universal set and A, B, C are three subsets of S, where A={a,c,d,f}, B ∩C={a,b,f}, find 
a) (A∪B)∩(A∪C)
b) B' ∪ C'

16) If U={1,2,3,4,5,6} be the universal set and A, B, C are three subsets of U where A{1,2,3,4} and B∪C={1,3,5,6}, find
a) (A∩B) ∪(A∩C)
b) B' ∩ C'

17) If A={x: -1≤ x ≤ 2} and B={x:0 < x ≤4}, find
a) A ∪ B
b) A∩ B
c) A - B
d) (A∪B) -(A∩B).

18) List the sets A, B and C, given that,
A∪B= ={p, q, r,s}, A∪ C={q,r,s,t}, A∩B={q,r}, A∩C={q,s}

19) Let A{1,2,3,4,5,6,7,8,9}, B={2,4,6,8}, C={1,3,5,7, 9}, D={3,4,5} and E={3,5}
Which set can equal X, if we are given the following information ?
a) X and B are disjoint
b) X ⊂ A but X ⊄ C
c) X ⊂ A but X ⊄ B
d) X ⊂ A but X ⊄ A

20) Applying set operation , find the HCF of the three numbers 15, 40, 105.      5

21) Using set operations find the LCM of the three numbers 12, 15 and 20.     60

22) Applying set operations , prove that, 2 + 3 = 5.

23) In an examination, 45% of the candidate have passed in English, 40% have passed in Bengali, while 30% have passed in both the subjects . Find the total number of candidates if 90 of them have failed in both the subjects.       200

24) It is known that in a group of people, each of whom speaks at least one of the languages English, Hindi and Bengali; 31 speak English, 36 speak Hindi and 27 speak Bengali . Ten speak both English and Hindi, 9 both English and Bengali, 11 both Hindi and Bengali, Prove that the group contains atleast 64 people and not more than 73 people .     

25) In a survey concerning the smoking habits of consumers, it was found that 50% smoke cigarette A, 45% smoke B, 40% smoke C, 25% smoke A and B, 10% smoke B and C, 16% smoke C and A, 8% smoke all the three brands. What percentage 
a) do not smoke.                 8%
b) smoke only A brand.               17%
c) smoke exactly two brands of cigarettes ?      27%

26) A factory inspector examined the defects in hardness, finish and dimensions of an item . After examining 100 items he gave the following report : 
All three defects 5, defect in hardness and finish 10, defect in dimension and finish 8, defect in dimensions and hardness 20. Defect in finish 30, in hardness 23 and in dimensions 50. The inspector was fined , why ?

     

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