প্ৰশ্নাৱলী(EXERCISE)
1. Express the following sets by the rule method.
(i) P = {2 , 4 , 6, 8 ,....}
(ii) Q = {.... -4, -2, 0, 2, 4, ....}
(iii) R = ф
Solution:
(i) P = {x /x is an set of even number }
P = {x /x এটা যুগ্ম সংখ্যাৰ সংহতি }
(ii) Q = {x / x is a set of even integer }
Q = {x/x এটা যুগ্ম অখণ্ড সংখ্যাৰ সংহতি }
(iii) R = { x / x is a void set }
Q = { x/x এটা ৰিক্ত সংহতি}
2. Express the elements of the following sets by Roster Method:
(i) N = {x/x is a natural number}
(ii) A = {a /a є N , a ≤ 6 }
(iii) V = {a : a is vowel of the English Alphabet}
Solution:
(i) N = { 1, 2, 3, 4, ... }
(ii) A = { 1, 2, 3, 4, 5, 6}
(iii) V = { a, e, i, o u}
3. State in words:
(i) A = { x / x has Property P}
(ii) W = {x /x is a number of days in a week }
(iii) P = { x / x ε N }
(iv) P = { x / x is an Indian Woman President}
Ans:
(i)
(ii) W = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday , Saturday}
(iii) P = { 1, 2, 3, 4,....}
(iv) W = { Pratibha Dev isingh Patil, Droupadi Murmu }
4. Rewrite the following statements using Set Notation.
(i) a does not belong to A.
(ii) p is not a proper subset of Q
(iii) x is a member of V.
Ans:
(i) a ∉ A
(ii) p ⊄ Q
(iii) x є V
5. Find the power set for the following:
(i) P = {1, 2, 3, 4}
(ii) Q = {2,4, 6}
(iii) R = {3,7, 8, 9}
(iv) S = ф
Solution:
(i) P = { ф , {1}, {2} {3}, {4}, {1,2 } {1,3 },{1,4 },{2,3 },{2,4}, {3,4}, {1,2,3}, {1,2,4},{1,3,4},{2,3,4}, P }
(ii) Q = { ф , {2}, {4} {6}, {2,4}, {2,6 } {4,6 }, Q }
(iii) R= { ф , {3}, {7} {8}, {9}, {3,7 } {3,8 },{3,9 },{7,8 },{7,9}, {8,9}, {3,7,8}, {3,7,9},{3,8,9},{7,8,9}, R }
(iv) S= ф
6. If U = {1,2,3,4,5,7}, A = {1, 2, 3, 4,}, B = {4,5,6} , C = {2,3,4}, then verify the following:
(i) A U ( B U C ) = ( A U B ) U C
(ii) A U ( B ⋂ C) = (A U B ) ⋂ ( A U C )
(iii) A ⋂ ( B ⋂ C ) = ( A ⋂ B ) ⋂ C
(iv) C ⋂ Cc= ф
(v) (Ac) = A
Ans: Given,
U = {1,2,3,4,5,7}
A = {1, 2, 3, 4,}
B = {4,5,6}
C = {2,3,4}
(i) A U ( B U C ) = ( A U B ) U C
LHS,
A U ( B U C )
= {1, 2, 3, 4,} U [ {4,5,6} U {2,3,4} ]
= {1, 2, 3, 4,} U {2,3, 4, 5, 6}
= { 1, 2, 3, 4, 5, 6 }
RHS,
( A U B ) U C
= [{1, 2, 3, 4,} U {4,5,6} ] U {2,3,4}
= {1, 2, 3, 4, 5, 6 } U {2, 3, 4 }
= {1, 2, 3, 4, 5, 6 }
So , LHS = RHS //
(ii) A U ( B ⋂ C) = (A U B ) ⋂ ( A U C )
LHS,
{1, 2, 3, 4,} U [{4,5,6}⋂{2,3,4}]
= {1, 2, 3, 4,} U {4 }
={1,2,3,4,}
RHS,
(A U B ) ⋂ ( A U C )
= [{1, 2, 3, 4,} U {4,5,6} ] ⋂ [{1, 2, 3, 4,} U{2,3,4}]
= {1,2,3,4,5,6} ⋂ {1,2,3,4}
= {1,2,3,4 }
So , LHS = RHS //
(iii) A ⋂ ( B ⋂ C ) = ( A ⋂ B ) ⋂ C
LHS,
A ⋂ ( B ⋂ C )
= {1, 2, 3, 4,} ⋂ [{4,5,6}⋂{2,3,4}]
= {1, 2, 3, 4,} ⋂ {2,3,4,5,6}
= {2,3,4}
RHS,
( A ⋂ B ) ⋂ C
= [{1, 2, 3, 4,}⋂ {4,5,6}] ⋂ {2,3,4}
= {4} ⋂ {2,3,4}
= {2,3,4}
So, LHS = RHS //
(iv) C ⋂ Cc= ф
={2,3,4} ⋂ ( U - C)
LHS,
= {2,3,4} ⋂ [{1,2,3,4,5,7} - {2,3,4} ]
= {2,3,4} ⋂ {1, 5,7}
= ф , RHS
(v) (Ac) = A
LHS,
(Ac) = U - A
= {1,2,3,4,5,7} - {1, 2, 3, 4,}
= {1,2,3,4}
= A
7. If U = {1,2, 3, 4, 5 } , A = { 1, 2, 4 }, B = { 2, 4, 5 }, then find the following -
(i) A U B
(ii) A ⋂ B
(iii) A - B
(iv) B - A
(v) Bᶜ ⋂ A
(vi) Aᶜ or A'
Solution: Given,
U = {1,2, 3, 4, 5 }
A = { 1, 2, 4 }
B = { 2, 4, 5 }
(i) A U B
= { 1, 2, 4 } U { 2, 4, 5 }
= {1,2,4,5 }
(ii) A ⋂ B
= { 1, 2, 4 } ⋂ { 2, 4, 5 }
= { 4 }
(iii) A - B
= { 1, 2, 4 } - { 2, 4, 5 }
= {1 }
(iv) B - A
= { 2, 4, 5 } - { 1, 2, 4 }
= { 2, 5 }
(v) Bᶜ ⋂ A
= (U - B) ⋂ A
= [{1,2, 3, 4, 5 } - { 2, 4, 5 }] ⋂ { 1, 2, 4 }
= {1,3,} ⋂ {1, 2, 4 }
= { 1 }
(vi) Aᶜ or A'
= U - A
= {1,2, 3, 4, 5 } - { 1, 2, 4 }
= { 3, 5 }
8. Distinguish between equal and equivalent sets.
Ans: (ক) সদৃশ সংহতিত যিকোনো দুটা সংহতিৰ উপাদানৰ সংখ্যা সমান হোৱাৰ লগতে উপাদানবোৰ একেই হয় ৷ আনহাতে সমতুল্য সংহতিত যিকোনা দুটা সংহতিৰ উপাদানবোৰ সমান হয় কিন্তু উপাদানবোৰ একে নহবও পাৰে ৷
(খ) সদৃশ সংহতি সমতুল্য হ'ব পাৰে, কিন্তু সমতুল্য সংহতি সদৃশ হবও পাৰে বা নহবও পাৰে ৷
উদাহৰণ -
ধৰাহ'ল ,
A = {1, 2, 3, 4,} , B = { 4, 3, 2, 1 } , C = { a , b, c, d }
ইয়াত ,
A = B ( সদৃশ সংহতি হয় ) আৰু ( সমতুল্য সংহতি হয় )
A ≠ C ( সদৃশ সংহতি নহয় )
A ≡ C ( সমতুল্য সংহতি হয় )
9. Find P x Q and Q x P
(i) P = { 2, 3 } , Q = { 4, 5, 6 }
(ii) P = { 1, 2, 3 }, Q = { 1, 3 }
Solution:
(i) P = { 2, 3 } , Q = { 4, 5, 6 }
P x Q = { 2, 3 } x { 4, 5, 6 }
= { (2, 4 ), (2 ,5 ), (2, 6), ( 3, 4 ), ( 3, 5 ), ( 3, 6) }
Q x P = { 4, 5, 6 } x { 2, 3 }
= { ( 4, 2 ), (4 , 3 ), (5, 2 ), ( 5, 3 ), ( 6, 2 ), ( 6 ,3) }
(ii) P = { 1, 2, 3 }, Q = { 1, 3 }
P x Q = { 1, 2, 3 } x { 1, 3 }
= { ( 1 , 1 ), ( 1, 3), ( 2, 1 ), ( 2, 3 ), ( 3, 1 ) , (3, 3 ) }
10. Express the following sets in Venn diagrams :
(i) If P = { a, b, c, d } and Q = { b, d, g, h }, then P U Q = ? and P ⋂ Q = ?
(ii) P = {1, 2, 3, 4, 5 } and Q = { 1, 3, 6, 7 } then P - Q = ? and Q - P = ?
Solution:
(i)
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Solution:
Here,
n(A) = 6
n(B) = 14
n(A U B) = 14
Now,
n(A U B) = n(A) + n(B) - n (A ⋂ B)
=> 14 = 6 + 14 - n (A ⋂ B)
=> 14 = 20 - n (A ⋂ B)
=> 14 - 20 = - n (A ⋂ B)
=> - 6 = - n (A ⋂ B)
=> n (A ⋂ B) = 6
∴ n (A ⋂ B) = 6 //
Previous Question - January/February, 2022
Economics ( Core)
Paper: C -2
(a) A U A' = ?
(i) U
(ii) A
(iii) A'
(iv) Ф
(b) The set of prime numbers is
(i) { 10, 12, 20 }
(ii) { 2, 3, 5 }
(iii) { 4, 6, 8}
(iv) None of the above.
2. (a) Define the following with examples:
Null set, Finite set , Equal set , Power set
3. (a) (i) Given A = { 4, 5, 6 }, B = { 3, 4, 6, 7 } and C = { 2 , 3, 6 }. Verify the distributive law of set operations.
(ii) The identity cards of 115 people were examined. Some of them were producing voter ID, some of had passport and some people were having both of them. If 65 people were having passports and 30 people had both, how many people have had voter ID only ?
(iii) If A = { 2, 3, 4, 5 } and B = { a, e, i, o, u }, then find A x B and B x A.
OR
(b) (i) What are whole numbers and rational numbers ? Give example.
(ii) If A = { 1, 3, 5 }, B = { 2, 4, 6, 8 }, C = { 2, 5, 10 }, U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
then verify that (A ⋂ B)' = A' U B'
(iii) In case of ordered pair ( a , b ) ≠ ( b, a ) unless a = b. Explain.
Solution:
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