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exerciseNo1.-Sets-and-Venn-Diagram

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1. Which of the following are well-defined sets?
.
(a) All the colors in the rainbow.
(b) All the points that lie on a straight line
(c)
All the honest members in the family.
(d) All the consonants of the English alphabet.
(e) All the tall boys of the school.
(f)
All the efficient doctors of the hospital.
(g) All the hardworking teachers in a school.
(h) All the prime numbers less than 100.
(i)
All the letters in the word GEOMETRY.
2.
Let A = (a, b, c, d, e, f}. Insert the appropriate symbol ∈ or ∉ in
the blank space.
(a) d __ A
(b) y __ A
(c)
m __ A
(d) a __ A
(e) e __ A
(f)
x __ A
3. Write the following sets in the set builder form.
(a) A = {2, 4, 6, 8}
(b) B = {3, 9, 27, 81}
(c)
C = {1, 4, 9, 16, 25}
(d)
D = {1, 3, 5, ......}
(e)
E = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ......., 52}
(f)
F = {-10, ......, -3, -2, -1, 0, 1, 2, …..., 5}
(g)
G = {O}
(h)
P={}
(i)
H = {-5, 5}
(j)
Q = {V, I, B, G, Y, 0, R}
4. Write the following sets in the roster form.
(a)
A = {x : x ∈ W, x ≤ 5}
(b) B = {x : x ∈ I, -3 < x < 3)
(c) C = {x : x is divisible by 12}
(d) D = {x : x = 3p, p ∈ W, p ≤ 3}
(e) E = {x : x = a2, a ∈ N, 3 < a < 7}
(f) F = {x : x = n/(n + 1), n ∈ N and n ≤ 4}
(g) G = {x : x ∈ N, 3x - 2 < 5}
(h) J = {x : x ∈ N, x2 < 16}
(i) K = {x : x is a prime number which is a divisor of 42}
(j) H = {x : x is a 2-digit natural number such that the sum of its
digits is 5}
5. Which of the following are the examples of an empty set?
(a) The set of even natural numbers divisible by 3.
(b) The set of all prime numbers divisible by 2.
(c) {x : x ∈ N, 5 < x < 6}
(d) The set of odd natural numbers divisible by 2.
(e) B = {O}
(f) C = { }
(g) D = {x : x ∈ R, x2 = -1}
(h) E = {x : x ∈ W, 3x + 1 = 2}
(i) P = {x : x is a prime number, 54 < x < 58}
(j) Q = {x : x = 2n + 3, n ∈ W, n ≤ 5}
6. Classify the following as finite and infinite sets.
(a)
The set of days in a week
(b) A = {x : x ∈ N x > 1}
(c) B = {x : x is an even prime number}
(d) C = {x : x is a multiple of 5}
(e) D = {x : x is a factor of 30}
(f) P = {x : x ∈ Z, x < -1}
(g) The set of all letters in the English alphabet
(h) The set of all real numbers
7. From the sets given below, identify the equal sets.
A = {3, 5, 9, 11}
Q = {m, s, t}
B = {8, 9, 1, 13}
R = {o, p, a, z}
C = {-3, 3}
T = {1, 8, 9, 13}
D = {s, t, m}
M = {3, -3}
P = {9, 3, 5, 11}
X = {a, o, z, p}
8. Are the following pairs of sets equal?
(a) A = {2}
B = {x : x ∈ N, x is an even prime number}.
(b) P = {1, 4, 9} Q = {x : x = n2, n ∈ N, n ≤ 3)
(c) X = {x : x ∈ W, x < 5}
Y = {x : x ∈ N, x ≤ 5}
(d) M = {a, b, c, d}
N = {p, q, r, s}
(e) D = {x : x is a multiple of 30}
E = {x : x is a factor of 10}
9. Which of the following are equivalent sets?
(a) A = {1, 2, 3}
B = {4, 5}
(b) P = {q, s, m}
Q = {6, 9, 12}
(c) X = {x : x is a prime number less than 10} Y = {x : x ∈ N, x ≤ 4}
(d) R = {x : x = 2n + 3, n < 4, n ∈ N} S = {x : x = n/(n + 1), n ∈ R,
n ≤ 4}
(e) The set of vowels in the English alphabet
(f) The set of consonants in the English alphabet
10. Find the cardinal number of the following sets.
(a) A = {x : x ∈ I, 2 < x < 7}
(b) B = {x : n ∈ N, x = n2, n < 3}
(c) The set of months in a year
(d) C = {x : x ∈ Z+, x < 100}
(e) D = {x : x = n3, n ∈ W, n < 5}
(f) The set of letters in the word MALAYALAM
11.
State whether true or false:
a. {x | x is a vowel in the word ‘EQUATION’} = {x | x is a vowel in
the word ‘EDUCATION’}
b. Collection of vowels used in the word ‘ALLAHABAD’ forms a set.
c. If X is the set of letters in the word ‘ROOM’; then X = {r, o, m}
d. If M is the set of letters used in the word ‘KOLKATA’; then M = {k,
o, l, a t}.
12. Write the set containing:
a. First six counting numbers
b. Three types of angles
c. Name of any three Prime Ministers of India.
d. Three types of triangles
e. Last six consonants of English alphabets.
f. First four vowels of English alphabets.
13. Write each of the following sets in the shortest possible way:
a.
{2, 7, 7, 2, 3, 7, 8}
b.
{10 - 5, 20 - 15, 30 - 25, 40 - 35, 37 - 32}
{2 + 8, 3 + 7, 4 + 6, 5 + 5, 6 + 4, 7 + 3}
14. Write the set of letters used in the word ‘GOOGLE’.
a. Write the set of vowels used in the word ‘BENGAL’.
b. Write the set of consonants used in the word ‘HONG-KONG’.
15. Write each of the following sets in Roster (tabular) Form and
also in Set-Builder Form:
a. Set of all natural numbers which can divide 24 completely.
b. Set of odd numbers between 20 and 35.
c.
d.
e.
f.
g.
h.
16.
Set of even natural numbers less than 25.
Set of letters used in the word ‘MASSACHUSETTS’.
Set of names of the first five months of a year.
Set of all two digit numbers which are perfect square also.
Set of letters used in the word ‘EDUCATION’.
Set of natural numbers divisible by 7
Write
a.
b.
c.
d.
17.
Let
down the union and intersection of the following pairs of sets:
A = {1, 2, 3, 4, 5, 6} B = {1, 3, 5, 7, 9}
X = {a, b, c, d, e}
Y = {c, e, f, g}
P = {x : x is a multiple of 2 between 9 and 21}
Q = {x : x is a multiple of 3 between 10 and 20}
M = {letters in the word ‘COMPUTER’}
N = {letters in the word ‘CALCULATOR’}
A = set of natural numbers less than 8,
B = {even natural numbers less than 12}
C = {Multiples of 3 between 5 and 15}
D = {Multiples of 4 greater than 6 and less than 20};
Find:
(a) B ∪ C
(b) A ∪ D
(c) C ∪ D
(d) A ∩ C
(e) B ∩ C) ∪ A
(f)
(D ∪ A) ∩ B
(g) (A ∩ C) ∪ (B ∩ D)
(h) (B ∪ D) ∩ (C ∪ A)
18.
If A {5, 7, 8, 9}, B = {3, 4, 5, 6} and C = {2, 4, 6, 8, 10};
Find:
(a)
n(A) + n(B)
(b)
n(A ∪ B)
(c)
n(A ∩ B)
(d)
n(A ∪ B) + n(A ∩ B)
(e)
n(B ∪ C)
(f)
n(B) + n(C) – n(B ∩ C)
(g)
Is n(A) + n(B) = n(A ∪ B) + n(A ∩ B)?
(h)
Is n(B ∪ C) = n(B) + n(C) - n(B ∩ C)?
19.
State, whether each of the following is true or false. In case, it is false,
write the correct answer.
a. If A = {0}, then n(A) = 0
b. n(∅) = 1
c. If T = {a, l, a, h, b, d, h}; then n(T) = 5
d. If B = {1, 5, 51, 15, 5, 1}; then n(B) = 6
20. Find the cardinal number of the following sets:
(a)
{}
(b)
{0}
(c)
{3, 7, 11, 15}
(d)
{3, 3, 3, 4, 4, 5}
(e)
{x : x is a letter in the word ‘STATISTICS’}
(f)
{x : x is an odd whole number less than 12}
(g)
{x : x ∈ N and x22 < 50}
(h)
{x : x is a factor of 12}
21. If O = {odd numbers less than 12} and E = {even numbers between 7
and 17}, show that:
n(O) – n(E) = 1.
22. Pairs of sets:
a. X = {letters of English alphabet upto ‘h’};
Y = {all the vowels of English alphabet}
b. A = {even numbers less than 10};
B = {odd numbers less than 10}
c. C = {multiple of 5 less than 30};
D = {multiple of 3 less than 20}
23.
If: A = Set of natural numbers,
B = Set of prime numbers and
C = Set of even prime numbers.
Draw Venn-diagram showing the relationship among the given sets A,
B and C.
24.
Let
M = {Natural numbers between 10 and 40; each divisible by 3}
N = {Natural numbers upto 40; each divisible by 4}.
a. Write each in roster form.
b. Draw a Venn-diagram showing the relationship between sets M
and set N.
25.
From the given Venn diagram, find the following sets:
(a)
A
(b)
B
(c)
A∪B
(d)
A∩B
26. Use the given Venn diagram to find the following sets:
(a)
X
(b)
Y
(c)
X∪Y
(d)
X∩Y
27. Find the sets from the disjoint sets:
a.
P
b.
Q
c.
P∪Q
d.
P∩Q
28. For each of the following Venn-diagrams, write the set P ∪ Q:
29. For each of the following Venn-diagrams, write the set X ∩ Y:
30.
31.
Let
A = Set of natural numbers,
B = Set of prime numbers and
C = Set of even prime numbers.
Draw Venn-diagram showing the relationship among the given
sets A, B and C.
Let M = {Natural numbers between 10 and 40; each divisible by 3}
N = {Natural numbers upto 40; each divisible by 4}.
a. Write each in roster form.
b. Draw a Venn-diagram showing the relationship between sets M and
set N.
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