Class 11 RD Sharma Solutions - Chapter 1 Sets - Exercise 1.4 | Set 2

The Sets are fundamental concepts in mathematics used to group and analyze collections of objects or elements. Chapter 1 of RD Sharma's Class 11 Mathematics textbook delves into the various properties and operations involving sets helping students understand how to manage collections in a structured way. In Exercise 1.4 | Set 2 students practice operations like union, intersection, and complement of sets through a series of problems.

What are Sets?

A set is a well-defined collection of distinct objects considered as an object in its own right. The objects in a set are called elements or members. The Sets are usually denoted by capital letters (A, B, C) and elements are written within curly braces. For example:

A={1,2,3,4}

Key operations on sets include:

• Union: Combining all elements from the two or more sets.
• Intersection: Common elements between the two or more sets.
• Complement: Elements not in the set but in the universal set.

Question 9. Write down all subsets of the following subsets: 

(i) {a}

(ii) {0,1}

(iii){a,b,c}

(iv){1,{1}}

(v) {ϕ}

Solution:

We know, if A is a set and B is a subset of A, then B is a subset of A, then B is called a proper subset of A if B ⊆ A and B≠A, ϕ, illustrated by B ⊆ A or B ⊂ A. 

For any set S with n elements, the power set has 2ⁿ elements. 

(i) Since n=1, the power set has 2¹ = 2 elements.The subsets are {a} and ϕ, but the set has no proper subsets. 

(ii) Since n=2, the power set has 2² = 4 elements. The elements of the power subset are ϕ, {0}, {1}, {0,1}.

(iii)Since n=3, the power set has 2³ = 8 elements.The elements of the power subset are ϕ, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}.

(iv) Since n=2, the power set has 2² = 4 elements. The elements of the power subset are ϕ, {1}, {{1}}, {1,{1}}.

(v)Since n=1, the power set has 2¹ = 2 elements, which are ϕ and {ϕ}.

Question 10. Write down all proper subsets of the following subsets: 

(i) {1,2}

(ii) {1,2,3}

(iii){1}

Solution: 

We know, if A is a set and B is a subset of A, then B is a subset of A, then B is called a proper subset of A if B ⊆ A and B≠A, ϕ, illustrated by B ⊆ A or B ⊂ A.

(i) The proper subsets are {1}, {2}

(ii) The proper subsets are {1}, {2}, {3}, {1,3}, {2,3}, {1,2}

(iii) The subsets are {1} and ϕ, but the set has no proper subsets. 

Question 11. What is the total number of proper subsets of a set consisting of n elements?

Solution

We know that, for any finite set A, having n elements, the total number of subsets of A has 2ⁿ elements. However, the number of proper subsets is 2ⁿ - 1, where A is not included. 

Question 12. If A is any set, prove that A ⊆ ∅ <=> A = ∅

Solution:

In order to prove that two sets A and B are equal, we need to show the following:

A ⊆ B and B ⊆ A.

Since, ∅ is a subset of every set, therefore, A ⊆ ∅.

Therefore, ∅ ⊆ A.

Hence, A = ∅

Let us assume now, that A =∅

Now, every set is a subset of itself, 

∅ = A ⊆ ∅ 

Hence, proved.  

Question 13. Prove that A ⊆ B, B ⊆ C, and C ⊆ A => A = C

Solution:

We have, A ⊆ B, B ⊆ C and C ⊆ A, therefore, A ⊆ B ⊆ C ⊆ A. 

Now, A is a subset of B and B is a subset of C,

So A is a subset of C, that is A ⊆ C.

Also, it is given, C ⊆ A.

We know, if A ⊆ C and C ⊆ A => A = C. 

Hence, proved. 

Question 14. How many elements has P(A)=?, if A = ∅.

Solution:

An empty set has zero elements. 

Therefore, size of A = n = 0. 

Power set of A P(A) = 2ⁿ = 2⁰ = 1 element. 

Question 15. What universal set(s) would you propose for the following? 

(i) The set of right triangles.

(ii) The set of isosceles triangles. 

Solution:

(i) The right triangle is a subset of the triangles. Therefore, the set of right triangles is a subset of set of triangles. 

=> Set of right triangles ⊆ Set of triangles, which becomes Universal set (U) in this case. 

(ii) The isosceles triangle is a special case of the triangles where two sides are equal. Therefore, the set of isosceles triangles is a subset of set of triangles.

=> Set of isosceles triangles ⊆ Set of triangles, which becomes Universal set (U) in this case. 

Question 16. If X= {8ⁿ - 7n -1 : n ∈ N} and Y = {49(n-1): n ∈ N}. Prove that X ⊆ Y.

Solution:

To prove X ⊆ Y, we need to show that each element of X belongs to Y. 

We have, 

X= {8ⁿ - 7n -1 : n ∈ N} 

Y = {49(n-1): n ∈ N}

So, let x ∈ X => x = 8^m- 7m - 1  for some m ∈ N

=> x = (1 + 7 )^m - 7m - 1

= {n \choose x}

= 

=({m \choose 0}1^m + {m \choose 1}1^{m-1} 7 +....+ {m \choose m}7^{m}) - 7m -1 

=49({m \choose 2}7^2 + {m \choose 3}7^{3} +...+ {m \choose m}7^{m-2}), m >=2 

= 49t_m, m >=2 where t_m = {m \choose 2}7^2 + {m \choose 3}7^{3} +...+ {m \choose m}7^{m-2}

For m = 1, we have, 

X = 8 - 7 x 1 - 1

   = 0

Hence, X contains all positive integral multiples of 49. 

Hence, Y contains all positive integral multiples of 49 and 0, for n =1.

Therefore, 

 X ⊆ Y.

Read More:

• What is Set Theory? Definition, Types, Operations
• Types Of Sets

Summary

Exercise 1.4 Set 2 covers advanced set operations, including intersection, union, and complement. Students learn to find the intersection, union, and complement of two sets, and understand the properties of these operations. Understanding advanced set operations is crucial for advanced math concepts and real-world applications.