Important Questions: Sets

# Important Questions: Sets | Mathematics (Maths) Class 11 - Commerce PDF Download

Q1: Write the following sets in roster form:
(i) A = {x : x is an integer and –3 ≤ x < 7}
Ans:
A = {x : x is an integer and –3 ≤ x < 7}
Integers are …-5, -4, -3, -2, -2, 0, 1, 2, 3, 4, 5, 6, 7, 8,…..
A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}

(ii) B = {x : x is a natural number less than 6}
Ans:
B = {x : x is a natural number less than 6}
Natural numbers are 1, 2, 3, 4, 5, 6, 7, ……
B = {1, 2, 3, 4, 5}

Q2: Let X = {1, 2, 3, 4, 5, 6}. If n represent any member of X, express the following as sets:
(i) n ∈ X but 2n ∉ X
Ans:
For X = {1, 2, 3, 4, 5, 6}, it is given that n ∈ X, but 2n ∉ X.
Let, A = {x | x ∈ X and 2x ∉ X}
Now, 1 ∉ A as 2.1 = 2 ∈ X
2 ∉ A as 2.2 = 4 ∈ X
3 ∉ A as 2.3 = 6 ∈ X
But 4 ∈ A as 2.4 = 8 ∉ X
5 ∈ A as 2.5 = 10 ∉ X
6 ∈ A as 2.6 = 12 ∉ X
Therefore, A = {4, 5, 6}

(ii) n + 5 = 8
Ans:
Let B = {x | x ∈ X and x + 5 = 8}
Here, B = {3} as x = 3 ∈ X and 3 + 5 = 8 and there is no other element belonging to X such that x + 5 = 8.

(iii) n is greater than 4
Ans:
Let C = {x | x ∈ X, x > 4}
Therefore, C = {5, 6}

Q3: Use the properties of sets to prove that for all the sets A and B, A – (A ∩ B) = A – B
Ans:
A – (A ∩ B) = A ∩ (A ∩ B)′ (since A – B = A ∩ B′)
= A ∩ (A′ ∪ B′) [by De Morgan’s law)
= (A∩A′) ∪ (A∩ B′) [by distributive law]
= φ ∪ (A ∩ B′)
= A ∩ B′ = A – B
Hence, proved that A – (A ∩ B) = A – B.

Q4: In a class of 60 students, 23 play hockey, 15 play basketball,20 play cricket and 7 play hockey and basketball, 5 play cricket and basketball, 4 play hockey and cricket, 15 do not play any of the three games. Find
(i) How many play hockey, basketball and cricket
(ii) How many play hockey but not cricket
(iii) How many play hockey and cricket but not basketball
Ans:
The Venn diagram of the given data is:

15 students do not play any of three games.
n(H ∪ B ∪ C) = 60 – 15 = 45
n(H ∪ B ∪ C) = n(H) + n(B) + n(C) – n(H ∩ B) – n(B ∩ C) – n(C ∩ H) + n(H ∩ B ∩ C)
45 = 23 + 15 + 20 – 7 – 5 – 4 + d
45 = 42 + d
d = 45- 42 = 3
Number of students who play all the three games = 3
Therefore, the number of students who play hockey, basketball and cricket = 3
a + d = 7
a = 7 – 3 = 4
b + d = 4
b = 4 – 3 = 1
a + b + d + e = 23
4 + 1 + 3 + e = 23
e = 15
Similarly, c = 2, g =14, f = 6
Number of students who play hockey but not cricket = a + e
= 4 + 15
= 19
Number of students who play hockey and cricket but not basketball = b = 1

Q5: In a survey of 600 students in a school, 150 students were found to be drinking Tea and 225 drinking Coffee, 100 were drinking both Tea and Coffee. Find how many students were drinking neither Tea nor Coffee.
Ans:
Given,
Total number of students = 600
Number of students who were drinking Tea = n(T) = 150
Number of students who were drinking Coffee = n(C) = 225
Number of students who were drinking both Tea and Coffee = n(T ∩ C) = 100
n(T U C) = n(T) + n(C) – n(T ∩ C)
= 150 + 225 -100
= 375 – 100
= 275
Hence, the number of students who are drinking neither Tea nor Coffee = 600 – 275 = 325

Q6: Write the following sets in the roster form.
(i) A = {x | x is a positive integer less than 10 and 2x – 1 is an odd number}
Ans:
2x – 1 is always an odd number for all positive integral values of x since 2x is an even number.
In particular, 2x – 1 is an odd number for x = 1, 2, … , 9.
Therefore, A = {1, 2, 3, 4, 5, 6, 7, 8, 9}

(ii) C = {x : x2 + 7x – 8 = 0, x ∈ R}
Ans: x2 + 7x – 8 = 0
(x + 8) (x – 1) = 0
x = – 8 or x = 1
Therefore, C = {– 8, 1}

Q7: Given that N = {1, 2, 3, …, 100}, then
(i) Write the subset A of N, whose elements are odd numbers.
Ans:
A = {x | x ∈ N and x is odd}
A = {1, 3, 5, 7, …, 99}

(ii) Write the subset B of N, whose elements are represented by x + 2, where x ∈ N.
Ans:
B = {y | y = x + 2, x ∈ N}
1 ∈ N, y = 1 + 2 = 3
2 ∈ N, y = 2 + 2 = 4, and so on.
Therefore, B = {3, 4, 5, 6, … , 100}

Q8: Let U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.
Find A′, B′, A′ ∩ B′, A ∪ B and hence show that ( A ∪ B )′ = A′∩ B′.
Ans:
Given,
U = {1, 2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}
A′ = {1, 4, 5, 6}
B′ = { 1, 2, 6 }.
Hence, A′ ∩ B′ = { 1, 6 }
Also, A ∪ B = { 2, 3, 4, 5 }
(A ∪ B)′ = { 1, 6 }
Therefore, ( A ∪ B )′ = { 1, 6 } = A′ ∩ B′

Q9: Let U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}, find
(i) A′ ∪ (B ∩ C′)
(ii) (B – A) ∪ (A – C)
Ans:
Given,
U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, B = {3, 5} and C = {1, 2, 4, 7}
(i) A′ = {1, 3, 5, 7}
C′ = {3, 5, 6}
B ∩ C′ = {3, 5}
A′ ∪ (B ∩ C′) = {1, 3, 5, 7}

(ii) B – A = {3, 5}
A – C = {6}
(B – A) ∪ (A – C) = {3, 5, 6}

Q10: Let U = {x : x ∈ N, x ≤ 9}; A = {x : x is an even number, 0 < x < 10}; B = {2, 3, 5, 7}. Write the set (A U B)’.
Ans:
Let U = {x : x ∈ N, x ≤ 9}; A = {x : x is an even number, 0 < x < 10}; B = {2, 3, 5, 7}
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {2, 4, 6, 8}
A U B = {2, 3, 4, 5, 6, 7, 8}
(A U B)’ = {1, 9}

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## Mathematics (Maths) Class 11

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## FAQs on Important Questions: Sets - Mathematics (Maths) Class 11 - Commerce

 1. What is a set in mathematics?
A set in mathematics is a well-defined collection of distinct objects, considered as an object in its own right. These objects can be anything, such as numbers, letters, or even other sets. The main defining feature of a set is that each object in the set is unique and there is no specific order or arrangement for the objects within the set.
 2. How are sets typically represented in mathematics?
Sets are commonly represented using curly braces {}. The objects within the set are listed inside the braces, separated by commas. For example, a set of even numbers less than 10 can be represented as {2, 4, 6, 8}.
 3. What is the cardinality of a set?
The cardinality of a set refers to the number of elements or objects in the set. It provides a measure of the "size" of the set. For example, the cardinality of the set {apple, banana, orange} is 3, as it contains three distinct fruits.
 4. What is the difference between a subset and a proper subset?
A subset is a set that contains all the elements of another set, including the possibility of being equal. On the other hand, a proper subset is a subset that contains all the elements of another set, but is not equal to that set. In other words, a proper subset is a subset that is strictly smaller than the original set.
 5. How are sets related to mathematical operations like union and intersection?
Sets can be combined or manipulated using various mathematical operations. The union of two sets is a new set that contains all the elements from both sets, without any duplicates. The intersection of two sets is a new set that contains only the elements that are common to both sets. These operations allow for the comparison and analysis of sets in different contexts.

## Mathematics (Maths) Class 11

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