OP Malhotra Class 11 Maths Solutions Chapter 1 Sets Exercise 1 (B)

S Chand Class 11 ICSE Maths Solutions Chapter 1 Sets Ex 1(b)

Question 1. Find the subsets of
(i) (a)
(ii) {Reena, Sonu}
(iii) \( \Phi \)
(iv) {5, {7}}
Answer:
(i) The subsets of \( \{a\} \) are \( \Phi, \{a\} \).
(ii) The subsets of \( \{\text{Reena, Sonu}\} \) are \( \Phi, \{\text{Sonu}\}, \{\text{Reena}\}, \{\text{Sonu, Reena}\} \).
(iii) The subsets of \( \Phi \) are \( \Phi \). This is because the empty set has only one subset, which is itself.
(iv) The subsets of \( \{5, \{7\}\} \) are \( \Phi, \{5\}, \{\{7\}\}, \{5, \{7\}\} \).
In simple words: A subset is a set formed by taking some or all elements from another set, or no elements at all. The empty set \( (\Phi) \) is always a subset of any set.

🎯 Exam Tip: Remember that if a set has 'n' elements, it will have \( 2^n \) subsets. For proper subsets, it is \( 2^n - 1 \).

Question 2. Let \( A = \{p, q, r\} \)
(i) List all the subsets of A.
(ii) List all the proper subsets of A.
Answer:
Given set \( A = \{p, q, r\} \). This set has 3 elements.
(i) The subsets of A are:
\( \Phi \)
\( \{p\}, \{q\}, \{r\} \)
\( \{p, q\}, \{q, r\}, \{p, r\} \)
\( \{p, q, r\} \)
There are \( 2^3 = 8 \) subsets in total.
(ii) The proper subsets of A are all the subsets except the set A itself. So, they are:
\( \Phi \)
\( \{p\}, \{q\}, \{r\} \)
\( \{p, q\}, \{q, r\}, \{p, r\} \)
There are \( 2^3 - 1 = 7 \) proper subsets. Proper subsets always have fewer elements than the original set.
In simple words: Subsets are all the possible small groups you can make from a set, including an empty group and the full group itself. Proper subsets are all these groups except the full group.

🎯 Exam Tip: Always list subsets systematically by number of elements (zero, one, two, etc.) to ensure none are missed. The empty set is always a subset and a proper subset (unless the original set is empty).

Question 3. Let \( P = \{\text{whole numbers less than 30}\} \)
(i) List the subset Q (even numbers)
(ii) List the subset R (odd numbers)
(iii) List the subset S (prime numbers)
(iv) List the subset T (square numbers)
(v) List the subset U (triangle numbers)
Answer:
First, we define the main set \( P \). Whole numbers less than 30 means numbers from 0 up to 29. So, \( P = \{0, 1, 2, 3, ..., 29\} \).
(i) Subset Q consists of even numbers from P:
\( Q = \{0, 2, 4, 6, ..., 28\} \)
(ii) Subset R consists of odd numbers from P:
\( R = \{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29\} \)
(iii) Subset S consists of prime numbers from P (numbers only divisible by 1 and themselves):
\( S = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\} \)
(iv) Subset T consists of square numbers from P (numbers obtained by squaring a whole number):
\( T = \{0^2, 1^2, 2^2, 3^2, 4^2, 5^2\} = \{0, 1, 4, 9, 16, 25\} \)
(v) Subset U consists of triangle numbers from P (numbers formed by adding a sequence of natural numbers, or using the formula \( T_n = \frac{n(n+1)}{2} \)):
\( U = \{0, 1, 3, 6, 10, 15, 21, 28\} \)
In simple words: We start with all numbers from 0 to 29. Then, we pick out specific types of numbers to make smaller sets: even numbers, odd numbers, prime numbers, square numbers, and triangle numbers.

🎯 Exam Tip: When defining sets based on properties, ensure you include all elements that fit the description and exclude any that do not, especially when dealing with specific types like prime or triangle numbers.

Question 4. Tell in each of the following, whether first set is a subset of the second set or not.
(i) A = Set of letters in the word 'LATE'
B = Set of letters in the word 'PLATE'
(ii) P = Set of even prime numbers.
Q = \( \{x \mid x = 2p, p \in \mathbb{N} \text{ and } 1 \leq p \leq 3\} \)
(iii) L = Set of digits in the number 1590
M = Set of digits in the number 178902
(iv) E = Set of all triangles having 4 sides.
F = Set of digits in the number '100'.
Answer:
(i) First, we list the elements for sets A and B:
Set A = \( \{L, A, T, E\} \)
Set B = \( \{P, L, A, T, E\} \)
Every element in set A (L, A, T, E) is also present in set B. Therefore, A is a subset of B.
(ii) First, define sets P and Q:
Set P = \( \{\text{even prime numbers}\} = \{2\} \) (since 2 is the only even prime number).
For set Q, we find the values of x for \( p \in \mathbb{N} \) (natural numbers) from 1 to 3:
When \( p = 1 \implies x = 2 \times 1 = 2 \)
When \( p = 2 \implies x = 2 \times 2 = 4 \)
When \( p = 3 \implies x = 2 \times 3 = 6 \)
So, set Q = \( \{2, 4, 6\} \).
Since every element of set P (which is just 2) is also an element of set Q, P is a subset of Q.
(iii) First, define sets L and M:
Set L = \( \{\text{digits in the number 1590}\} = \{1, 5, 9, 0\} \)
Set M = \( \{\text{digits in the number 178902}\} = \{1, 7, 8, 9, 0, 2\} \)
The digit 5 is in set L, but it is not in set M. Because of this, L is not a subset of M.
(iv) First, define sets E and F:
Set E = \( \{\text{all triangles having 4 sides}\} \). There are no triangles with 4 sides, so E is an empty set, \( E = \Phi \).
Set F = \( \{\text{digits in the number '100'}\} = \{1, 0\} \).
The empty set is always considered a subset of every set. Thus, E is a subset of F.
In simple words: We check if every item in the first set can also be found in the second set. If yes, it's a subset. If even one item from the first set is missing from the second, it's not a subset. The empty set is special; it's always a subset of any other set.

🎯 Exam Tip: When dealing with letters in words, ensure you list unique letters only. For conditions like \( p \in \mathbb{N} \), carefully substitute values to find all elements of the set.

Question 5. Write the proper subsets of the following sets :
(i) {7}
(ii) {1, 3}
(iii) {c, a, b}
(iv) \( \Phi \)
Answer:
(i) For the set \( \{7\} \), it has 1 element. The proper subsets are all subsets except the set itself. The only subset is \( \Phi \). Therefore, the only proper subset is \( \Phi \).
(ii) For the set \( \{1, 3\} \), it has 2 elements. The proper subsets are all subsets except \( \{1, 3\} \). They are: \( \Phi, \{1\}, \{3\} \).
(iii) For the set \( \{c, a, b\} \), it has 3 elements. The proper subsets are all subsets except \( \{c, a, b\} \). They are:
\( \Phi \)
\( \{c\}, \{a\}, \{b\} \)
\( \{c, a\}, \{a, b\}, \{c, b\} \)
(iv) For the set \( \Phi \) (empty set), it has 0 elements. The empty set does not have any proper subsets because its only subset is itself, and a proper subset must be different from the original set. Therefore, it has no proper subsets.
In simple words: A proper subset is a smaller group that is part of a main set, but it can't be the exact same as the main set. For the empty set, there are no smaller groups possible.

🎯 Exam Tip: Remember that a proper subset must contain fewer elements than the original set. The empty set is a proper subset of any non-empty set.

Question 6. How many subsets do the following sets have ?
(i) A set having 5 elements.
(ii) The set of letters of the word 'CENTENARY'
Answer:
(i) We use the formula that a set with \( n \) elements has \( 2^n \) subsets. For a set having 5 elements, the number of subsets is \( 2^5 \).
\( 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \).
So, a set with 5 elements has 32 subsets.
(ii) First, we need to find the unique letters in the word 'CENTENARY' to determine the number of elements in the set.
The letters in 'CENTENARY' are C, E, N, T, E, N, A, R, Y.
The unique letters (ignoring repeats) are C, E, N, T, A, R, Y.
So, the set of letters is \( \{C, E, N, T, A, R, Y\} \).
This set has 7 elements. Using the formula \( 2^n \), the number of subsets is \( 2^7 \).
\( 2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128 \).
Thus, the set of letters in 'CENTENARY' has 128 subsets.
In simple words: To find how many groups (subsets) you can make from a set, count its unique items (elements). Then, multiply 2 by itself that many times. For letters in a word, only count each different letter once.

🎯 Exam Tip: Always identify the unique elements in a set first to correctly determine 'n' (the number of elements), especially with words that have repeating letters.

Question 7. How many proper subsets do the following sets have ?
(i) The set of factors of 12.
(ii) The set \( A = \{x \mid x \text{ is a prime number and } x < 20\} \)
Answer:
We know that if a set has \( n \) elements, the number of proper subsets is given by the formula \( 2^n - 1 \).
(i) First, we find the factors of 12. These are the numbers that divide 12 evenly.
The factors of 12 are 1, 2, 3, 4, 6, 12.
So, set A = \( \{1, 2, 3, 4, 6, 12\} \). This set has 6 elements, so \( n=6 \).
The number of proper subsets is \( 2^n - 1 = 2^6 - 1 \).
\( 2^6 = 64 \).
So, \( 64 - 1 = 63 \).
This set has 63 proper subsets.
(ii) First, we list the elements for set A based on the given condition: prime numbers less than 20.
Set A = \( \{2, 3, 5, 7, 11, 13, 17, 19\} \). This set has 8 elements, so \( n=8 \).
The number of proper subsets is \( 2^n - 1 = 2^8 - 1 \).
\( 2^8 = 256 \).
So, \( 256 - 1 = 255 \).
This set has 255 proper subsets.
In simple words: To find the count of proper subsets, first list all the unique items in the set. Count how many items there are (let's call it 'n'). Then, multiply 2 by itself 'n' times, and subtract 1 from that answer.

🎯 Exam Tip: Make sure to correctly identify all elements of the set before calculating \( n \). For proper subsets, always subtract 1 from the total number of subsets.

Question 8. Answer true or false
(i) \( 3 \subseteq \{3, 0\} \)
(ii) \( \{3\} \subseteq \{3, 0\} \)
(iii) \( \Phi \subset \{3, 0\} \)
(iv) \( 0 \in \{3, 0\} \)
(v) \( \Phi \subset \{3, 0\} \)
(vi) Empty set is a subset of every set.
(vii) For any two sets A and B either \( A \subseteq B \) or \( B \subseteq A \).
(viii) Every set has a proper subset.
(ix) Every subset of a finite set is finite.
(x) Every subset of an infinite set is infinite.
Answer:
(i) The statement \( 3 \subseteq \{3, 0\} \) is false. The symbol \( \subseteq \) is used for subsets, not for elements. Here, 3 is an element of the set, written as \( 3 \in \{3, 0\} \).
(ii) The statement \( \{3\} \subseteq \{3, 0\} \) is true. The set \( \{3\} \) is a subset of \( \{3, 0\} \) because all its elements (only 3) are also in \( \{3, 0\} \).
(iii) The statement \( \Phi \subset \{3, 0\} \) is true. The empty set is a proper subset of any non-empty set. \( \{3, 0\} \) is a non-empty set.
(iv) The statement \( 0 \in \{3, 0\} \) is true. The number 0 is an element found within the set \( \{3, 0\} \).
(v) The statement \( \Phi \subset \{3, 0\} \) is true. This repeats the logic from (iii) and confirms that the empty set is a proper subset of \( \{3, 0\} \).
(vi) The statement "Empty set is a subset of every set" is true. This is a fundamental property of sets.
(vii) The statement "For any two sets A and B either \( A \subseteq B \) or \( B \subseteq A \)" is false. For example, if \( A = \{1, 2\} \) and \( B = \{3, 4\} \), neither is a subset of the other. Sets like these are called incomparable. If \( A = \{1,2\} \) and \( B = \{4,5\} \), then neither is a subset of the other.
(viii) The statement "Every set has a proper subset" is false. The empty set \( \Phi \) has only one subset (itself), and therefore has no proper subsets.
(ix) The statement "Every subset of a finite set is finite" is true. If a set has a limited number of elements, any group formed from those elements will also have a limited number of elements.
(x) The statement "Every subset of an infinite set is infinite" is false. An infinite set can have finite subsets. For instance, the set of natural numbers \( \mathbb{N} = \{1, 2, 3, ...\} \) is infinite, but \( \{1\} \) is a finite subset of \( \mathbb{N} \). The empty set \( \Phi \) is also a finite subset of any infinite set.
In simple words: We check if each sentence about sets is correct. Remember that an item is 'in' a set, while a smaller group (subset) is 'part of' a set. The empty set is special: it's a subset of everything, but has no proper subsets itself.

🎯 Exam Tip: Pay close attention to the symbols \( \in \) (element of), \( \subseteq \) (subset), and \( \subset \) (proper subset). A common mistake is confusing an element with a set containing that element.

Question 9. Find the power set of each of the following sets :
(i) A = {digits in the number 98}
(ii) B = {letters in the word 'KID'}
(iii) S = {2, 3}
(iv) T = {4, 7, 9}
Answer:
The power set of a set is the set of all possible subsets of that given set.
(i) First, define set A: Digits in the number 98 are 9 and 8. So, \( A = \{8, 9\} \).
The subsets of A are \( \Phi, \{8\}, \{9\}, \{8, 9\} \).
The power set of A, denoted \( P(A) \), is \( P(A) = \{\Phi, \{8\}, \{9\}, \{8, 9\}\} \).
(ii) First, define set B: Letters in the word 'KID' are K, I, D. So, \( B = \{K, I, D\} \).
The subsets of B are \( \Phi, \{K\}, \{I\}, \{D\}, \{K, I\}, \{K, D\}, \{I, D\}, \{K, I, D\} \).
The power set of B, denoted \( P(B) \), is \( P(B) = \{\Phi, \{K\}, \{I\}, \{D\}, \{K, I\}, \{K, D\}, \{I, D\}, \{K, I, D\}\} \).
(iii) First, define set S: \( S = \{2, 3\} \).
The subsets of S are \( \Phi, \{2\}, \{3\}, \{2, 3\} \).
The power set of S, denoted \( P(S) \), is \( P(S) = \{\Phi, \{2\}, \{3\}, \{2, 3\}\} \).
(iv) First, define set T: \( T = \{4, 7, 9\} \).
The subsets of T are \( \Phi, \{4\}, \{7\}, \{9\}, \{4, 7\}, \{4, 9\}, \{7, 9\}, \{4, 7, 9\} \).
The power set of T, denoted \( P(T) \), is \( P(T) = \{\Phi, \{4\}, \{7\}, \{9\}, \{4, 7\}, \{4, 9\}, \{7, 9\}, \{4, 7, 9\}\} \).
In simple words: To find the power set, list out every single possible subset you can make from the original set. Then, gather all these subsets together into a new big set; that's the power set.

🎯 Exam Tip: The number of elements in a power set is always \( 2^n \), where 'n' is the number of elements in the original set. This helps you check if you have listed all subsets correctly.