ML Aggarwal: Operation on Sets Venn Diagram - 1

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1. If A = {0, 1, 2, 3… 8}, B = {3, 5, 7, 9, 11} and C = {0, 5, 10, 20}, find 
(i) A ? B 
(ii) A ? C 
(iii) B ? C 
(iv) A n B 
(v) A n C 
(vi) B n C 
Also find the cardinal number of the sets B ? C, A ? B, A n C and B 
n C. 
Solution:- 
From the question it is given that, 
A = {0, 1, 2, 3, 4, 5, 6, 7, 8} 
B = {3, 5, 7, 9, 11} 
C = {0, 5, 10, 20} 
(i) A ? B 
A ? B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11} 
Cardinal number of set i.e. n(A ? B) = 11 
(ii) A ? C 
A ? C = {0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20} 
Cardinal number of set i.e. n(A ? C) = 11 
(iii) B ? C 
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1. If A = {0, 1, 2, 3… 8}, B = {3, 5, 7, 9, 11} and C = {0, 5, 10, 20}, find 
(i) A ? B 
(ii) A ? C 
(iii) B ? C 
(iv) A n B 
(v) A n C 
(vi) B n C 
Also find the cardinal number of the sets B ? C, A ? B, A n C and B 
n C. 
Solution:- 
From the question it is given that, 
A = {0, 1, 2, 3, 4, 5, 6, 7, 8} 
B = {3, 5, 7, 9, 11} 
C = {0, 5, 10, 20} 
(i) A ? B 
A ? B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11} 
Cardinal number of set i.e. n(A ? B) = 11 
(ii) A ? C 
A ? C = {0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20} 
Cardinal number of set i.e. n(A ? C) = 11 
(iii) B ? C 
B ? C = {0, 3, 5, 7, 9, 10, 11, 20} 
Cardinal number of set i.e. n(B ? C) = 8 
(iv) A n B 
A n B = {3, 5, 7} 
Cardinal number of set i.e. n(A n B) = 3 
(v) A n C 
A n C = {0, 5} 
Cardinal number of set i.e. n(A n C) = 2 
(vi) B n C 
B n C = {5} 
Cardinal number of set i.e. n(B n C) = 1 
 
2. Find A’ when  
(i) A= {0, 1, 4, 7} and E, = {x | x ? W, x = 10} 
Solution:- 
From the question, 
A = {0, 1, 4, 7} 
E = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 
Then, A’ = {2, 3, 5, 6, 8, 9, 10} 
 
(ii) A = {consonants} and ? = {alphabets of English} 
Solution:- 
A = {consonants} 
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1. If A = {0, 1, 2, 3… 8}, B = {3, 5, 7, 9, 11} and C = {0, 5, 10, 20}, find 
(i) A ? B 
(ii) A ? C 
(iii) B ? C 
(iv) A n B 
(v) A n C 
(vi) B n C 
Also find the cardinal number of the sets B ? C, A ? B, A n C and B 
n C. 
Solution:- 
From the question it is given that, 
A = {0, 1, 2, 3, 4, 5, 6, 7, 8} 
B = {3, 5, 7, 9, 11} 
C = {0, 5, 10, 20} 
(i) A ? B 
A ? B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11} 
Cardinal number of set i.e. n(A ? B) = 11 
(ii) A ? C 
A ? C = {0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20} 
Cardinal number of set i.e. n(A ? C) = 11 
(iii) B ? C 
B ? C = {0, 3, 5, 7, 9, 10, 11, 20} 
Cardinal number of set i.e. n(B ? C) = 8 
(iv) A n B 
A n B = {3, 5, 7} 
Cardinal number of set i.e. n(A n B) = 3 
(v) A n C 
A n C = {0, 5} 
Cardinal number of set i.e. n(A n C) = 2 
(vi) B n C 
B n C = {5} 
Cardinal number of set i.e. n(B n C) = 1 
 
2. Find A’ when  
(i) A= {0, 1, 4, 7} and E, = {x | x ? W, x = 10} 
Solution:- 
From the question, 
A = {0, 1, 4, 7} 
E = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 
Then, A’ = {2, 3, 5, 6, 8, 9, 10} 
 
(ii) A = {consonants} and ? = {alphabets of English} 
Solution:- 
A = {consonants} 
? = {alphabets of English} 
A’ = {Vowels} 
 
(iii) A = {boys in class VIII of all schools in Bengaluru} and ? = 
{students in class VIII of all schools in Bengaluru} 
Solution:- 
A’ = {Girls in class VIII of all schools in Bengaluru} 
 
(iv) A = {letters of KALKA} and ? = {letters of KOLKATA} 
Solution:- 
From the question it is given that, 
A = {K, A, L, K, A} 
? = {K, O, L, K, A, T, A} 
A’ = {O, T} 
 
(v) A = {odd natural numbers} and ? = {whole numbers}. 
Solution:- 
From the question it is given that, 
A = {odd natural numbers} 
? = {whole numbers} 
A’ = {0, even whole numbers} 
3. If A {x : x ? N and 3 < x < 7} and B = {x : x ? W and x = 4}, find 
(i) A ? B 
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1. If A = {0, 1, 2, 3… 8}, B = {3, 5, 7, 9, 11} and C = {0, 5, 10, 20}, find 
(i) A ? B 
(ii) A ? C 
(iii) B ? C 
(iv) A n B 
(v) A n C 
(vi) B n C 
Also find the cardinal number of the sets B ? C, A ? B, A n C and B 
n C. 
Solution:- 
From the question it is given that, 
A = {0, 1, 2, 3, 4, 5, 6, 7, 8} 
B = {3, 5, 7, 9, 11} 
C = {0, 5, 10, 20} 
(i) A ? B 
A ? B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11} 
Cardinal number of set i.e. n(A ? B) = 11 
(ii) A ? C 
A ? C = {0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20} 
Cardinal number of set i.e. n(A ? C) = 11 
(iii) B ? C 
B ? C = {0, 3, 5, 7, 9, 10, 11, 20} 
Cardinal number of set i.e. n(B ? C) = 8 
(iv) A n B 
A n B = {3, 5, 7} 
Cardinal number of set i.e. n(A n B) = 3 
(v) A n C 
A n C = {0, 5} 
Cardinal number of set i.e. n(A n C) = 2 
(vi) B n C 
B n C = {5} 
Cardinal number of set i.e. n(B n C) = 1 
 
2. Find A’ when  
(i) A= {0, 1, 4, 7} and E, = {x | x ? W, x = 10} 
Solution:- 
From the question, 
A = {0, 1, 4, 7} 
E = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 
Then, A’ = {2, 3, 5, 6, 8, 9, 10} 
 
(ii) A = {consonants} and ? = {alphabets of English} 
Solution:- 
A = {consonants} 
? = {alphabets of English} 
A’ = {Vowels} 
 
(iii) A = {boys in class VIII of all schools in Bengaluru} and ? = 
{students in class VIII of all schools in Bengaluru} 
Solution:- 
A’ = {Girls in class VIII of all schools in Bengaluru} 
 
(iv) A = {letters of KALKA} and ? = {letters of KOLKATA} 
Solution:- 
From the question it is given that, 
A = {K, A, L, K, A} 
? = {K, O, L, K, A, T, A} 
A’ = {O, T} 
 
(v) A = {odd natural numbers} and ? = {whole numbers}. 
Solution:- 
From the question it is given that, 
A = {odd natural numbers} 
? = {whole numbers} 
A’ = {0, even whole numbers} 
3. If A {x : x ? N and 3 < x < 7} and B = {x : x ? W and x = 4}, find 
(i) A ? B 
(ii) A n B 
(iii) A – B 
(iv) B – A 
Solution:- 
From the question it is given that, 
A = {x : x ? N and 3 < x < 7} 
A = {4, 5, 6} 
B = {x : x ? W and x = 4} 
B = {0, 1, 2, 3, 4} 
Then, 
(i) A ? B 
A ? B = {0, 1, 2, 3, 4, 5, 6} 
(ii) A n B 
A n B = {4} 
(iii) A – B 
A – B = {5, 6} 
(iv) B – A 
B – A = {0, 1, 2, 3} 
 
4. If P = {x : x ? W and x < 6} and Q = {x : x ? N and 4 = x = 9}, find 
(i) P ? Q 
(ii) P n Q 
(iii) P – Q 
(iv) Q – P 
Is P ? Q a proper superset of P n Q ? 
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1. If A = {0, 1, 2, 3… 8}, B = {3, 5, 7, 9, 11} and C = {0, 5, 10, 20}, find 
(i) A ? B 
(ii) A ? C 
(iii) B ? C 
(iv) A n B 
(v) A n C 
(vi) B n C 
Also find the cardinal number of the sets B ? C, A ? B, A n C and B 
n C. 
Solution:- 
From the question it is given that, 
A = {0, 1, 2, 3, 4, 5, 6, 7, 8} 
B = {3, 5, 7, 9, 11} 
C = {0, 5, 10, 20} 
(i) A ? B 
A ? B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11} 
Cardinal number of set i.e. n(A ? B) = 11 
(ii) A ? C 
A ? C = {0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20} 
Cardinal number of set i.e. n(A ? C) = 11 
(iii) B ? C 
B ? C = {0, 3, 5, 7, 9, 10, 11, 20} 
Cardinal number of set i.e. n(B ? C) = 8 
(iv) A n B 
A n B = {3, 5, 7} 
Cardinal number of set i.e. n(A n B) = 3 
(v) A n C 
A n C = {0, 5} 
Cardinal number of set i.e. n(A n C) = 2 
(vi) B n C 
B n C = {5} 
Cardinal number of set i.e. n(B n C) = 1 
 
2. Find A’ when  
(i) A= {0, 1, 4, 7} and E, = {x | x ? W, x = 10} 
Solution:- 
From the question, 
A = {0, 1, 4, 7} 
E = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 
Then, A’ = {2, 3, 5, 6, 8, 9, 10} 
 
(ii) A = {consonants} and ? = {alphabets of English} 
Solution:- 
A = {consonants} 
? = {alphabets of English} 
A’ = {Vowels} 
 
(iii) A = {boys in class VIII of all schools in Bengaluru} and ? = 
{students in class VIII of all schools in Bengaluru} 
Solution:- 
A’ = {Girls in class VIII of all schools in Bengaluru} 
 
(iv) A = {letters of KALKA} and ? = {letters of KOLKATA} 
Solution:- 
From the question it is given that, 
A = {K, A, L, K, A} 
? = {K, O, L, K, A, T, A} 
A’ = {O, T} 
 
(v) A = {odd natural numbers} and ? = {whole numbers}. 
Solution:- 
From the question it is given that, 
A = {odd natural numbers} 
? = {whole numbers} 
A’ = {0, even whole numbers} 
3. If A {x : x ? N and 3 < x < 7} and B = {x : x ? W and x = 4}, find 
(i) A ? B 
(ii) A n B 
(iii) A – B 
(iv) B – A 
Solution:- 
From the question it is given that, 
A = {x : x ? N and 3 < x < 7} 
A = {4, 5, 6} 
B = {x : x ? W and x = 4} 
B = {0, 1, 2, 3, 4} 
Then, 
(i) A ? B 
A ? B = {0, 1, 2, 3, 4, 5, 6} 
(ii) A n B 
A n B = {4} 
(iii) A – B 
A – B = {5, 6} 
(iv) B – A 
B – A = {0, 1, 2, 3} 
 
4. If P = {x : x ? W and x < 6} and Q = {x : x ? N and 4 = x = 9}, find 
(i) P ? Q 
(ii) P n Q 
(iii) P – Q 
(iv) Q – P 
Is P ? Q a proper superset of P n Q ? 
Solution:- 
From the question it is given that, 
P = {x : x ? W and x < 6} 
P = {0, 1, 2, 3, 4, 5} 
Q = {x : x ? N and 4 = x = 9} 
Q = {4, 5, 6, 7, 8, 9} 
(i) P ? Q 
P ? Q = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 
(ii) P n Q 
P n Q = {4, 5} 
(iii) P – Q 
P – Q = {0, 1, 2, 3} 
(iv) Q – P 
Q – P = {6, 7, 8, 9} 
By observing the above sets, P ? Q is a proper superset of P n Q. 
 
5. If A = (letters of word INTEGRITY) and B = (letters of word 
RECKONING), find 
(i) A ? B 
(ii) A n B 
(iii) A – B 
(iv) B – A 
Also verify that: 
(a) n(A ? B) = n(A) + n(B) – n(A n B) 
(b) n(A – B) = n(A ? B) – n(B) = n(A) – n(A ? B)