RBSE (Rajasthan Board of Secondary Education) Class 11 Maths Chapter 1, “Sets,” covers fundamental concepts like the definition of sets, representation of sets, empty sets, finite and infinite sets, equal sets, subsets, power sets, universal sets, Venn diagrams, and operations on sets.
Table of Contents
Exercise 1.3 specifically focuses on subsets, power sets, and universal sets, along with understanding and using the symbols ⊂ (is a subset of), ⊆ (is a subset of or equal to), ⊂ (is not a subset of), and ∈ (is an element of).
Here are the typical types of questions found in Exercise 1.3, along with their solutions:
Key Concepts to Remember:
- Subset (⊂ or ⊆): Set A is a subset of set B if every element of A is also an element of B.
- A⊂B means A is a proper subset of B (A is a subset of B, but A is not equal to B).
- A⊆B means A is a subset of B (A can be equal to B).
- Element (∈): An element is a member of a set.
- Power Set (P(A)): The set of all possible subsets of a given set A. If a set A has n elements, then its power set P(A) has 2n elements.
- Universal Set (U): A set that contains all elements relevant to a particular context or problem.
Common Question Types and Solutions:
Question 1: Fill in the blanks with ⊂ or ⊂.
(i) {2,3,4}…{1,2,3,4,5} * Solution: {2,3,4}⊂{1,2,3,4,5} (Every element of the first set is in the second set).
(ii) {a,b,c}…{b,c,d} * Solution: {a,b,c}⊂{b,c,d} (Element ‘a’ is in the first set but not in the second).
(iii) {x:x is a student of Class XI of your school}…{x:x is a student of your school} * Solution: {x:x is a student of Class XI of your school}⊂{x:x is a student of your school} (All students of Class XI are students of the school).
(iv) {x:x is a circle in the plane}…{x:x is a circle in the same plane with radius 1 unit} * Solution: {x:x is a circle in the plane}⊂{x:x is a circle in the same plane with radius 1 unit} (A circle in the plane can have any radius, not just 1 unit).
(v) {x:x is a triangle in a plane}…{x:x is a rectangle in the plane} * Solution: {x:x is a triangle in a plane}⊂{x:x is a rectangle in the plane} (Triangles are not rectangles).
(vi) {x:x is an equilateral triangle in a plane}…{x:x is a triangle in the same plane} * Solution: {x:x is an equilateral triangle in a plane}⊂{x:x is a triangle in the same plane} (Every equilateral triangle is a type of triangle).
(vii) {x:x is an even natural number}…{x:x is an integer} * Solution: {x:x is an even natural number}⊂{x:x is an integer} (Even natural numbers are 2, 4, 6, … and these are all integers).
Question 2: Examine whether the following statements are true or false.
(i) {a,b}⊂{b,c,a} * Solution: False. Every element of {a,b} (i.e., ‘a’ and ‘b’) is present in {b,c,a}. So, {a,b}⊂{b,c,a} is true.
(ii) {a,e}⊂{x:x is a vowel in the English alphabet} * Solution: True. The set of vowels in the English alphabet is {a,e,i,o,u}. Since ‘a’ and ‘e’ are both present, the statement is true.
(iii) {1,2,3}⊂{1,3,5} * Solution: False. The element ‘2’ is in {1,2,3} but not in {1,3,5}.
(iv) {a}⊂{a,b,c} * Solution: True. The element ‘a’ is in both sets.
(v) {a}∈{a,b,c} * Solution: False. The elements of {a,b,c} are a, b, and c. The set {a} is not an element of {a,b,c}. It is a subset, not an element. (a∈{a,b,c} is true, but {a}∈{a,b,c} is false).
(vi) {x:x is an even natural number less than 6}⊂{x:x is a natural number which divides 36} * Solution: True. * {x:x is an even natural number less than 6}={2,4} * {x:x is a natural number which divides 36}={1,2,3,4,6,9,12,18,36} * Both 2 and 4 are present in the second set.
Question 3: Let A={1,2,{3,4},5}. Which of the following statements are incorrect and why?
(i) {3,4}⊂A * Solution: Incorrect. The element {3,4} is an element of set A, not a subset. For it to be a subset, its elements (3 and 4) would need to be directly in A, which they are not. Instead, {3,4} is treated as a single element within set A.
(ii) {3,4}∈A * Solution: Correct. {3,4} is indeed an element of set A.
(iii) {{3,4}}⊂A * Solution: Correct. The set containing the element {3,4} is a subset of A because {3,4} is an element of A.
(iv) 1∈A * Solution: Correct. 1 is an element of A.
(v) 1⊂A * Solution: Incorrect. 1 is an element, not a set. An element cannot be a subset.
(vi) {1,2,5}⊂A * Solution: Correct. Elements 1, 2, and 5 are all in A.
(vii) {1,2,5}∈A * Solution: Incorrect. The set {1,2,5} is not an element of A.
(viii) {1,2,3}⊂A * Solution: Incorrect. Element 3 is not directly in A. It is part of the element {3,4}.
(ix) Φ∈A * Solution: Incorrect. The empty set (Φ) is not an element of A.
(x) Φ⊂A * Solution: Correct. The empty set is a subset of every set.
(xi) {Φ}⊂A * Solution: Incorrect. For {Φ} to be a subset of A, Φ would need to be an element of A, which it is not.
Question 4: Write down all the subsets of the following sets:
(i) {a} * Solution: Φ,{a}
(ii) {a,b} * Solution: Φ,{a},{b},{a,b}
(iii) {1,2,3} * Solution: Φ,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}
(iv) Φ * Solution: Φ (The only subset of the empty set is the empty set itself).
Question 5: How many elements has P(A), if A=Φ?
- Solution: If A=Φ, then the number of elements in A, n(A)=0. The number of elements in the power set P(A) is 2n(A). So, 20=1. Therefore, P(A) has 1 element, which is {Φ}.
Question 6: Write the following as intervals:
(i) {x:x∈R,−4<x≤6} * Solution: (−4,6]
(ii) {x:x∈R,−12<x<−10} * Solution: (−12,−10)
(iii) {x:x∈R,0≤x<7} * Solution: [0,7)
(iv) {x:x∈R,3≤x≤4} * Solution: [3,4]
Question 7: Write the following intervals in set-builder form:
(i) (−3,0) * Solution: {x:x∈R,−3<x<0}
(ii) [6,12] * Solution: {x:x∈R,6≤x≤12}
(iii) (6,12] * Solution: {x:x∈R,6<x≤12}
(iv) [−23,5) * Solution: {x:x∈R,−23≤x<5}
Question 8: What universal set(s) would you propose for each of the following?
(i) The set of right triangles. * Solution: The universal set could be “The set of all triangles” or “The set of all polygons” or “The set of all points in a plane”. The most appropriate and specific universal set here would be “The set of all triangles”.
(ii) The set of isosceles triangles. * Solution: Similar to the above, the most appropriate universal set would be “The set of all triangles”.
Question 9: Given the sets A={1,3,5}, B={2,4,6} and C={0,2,4,6,8}, which of the following may be considered as universal set(s) for all the three sets A, B and C?
(i) {0,1,2,3,4,5,6} * Solution: No. Element ‘8’ from set C is not present in this set.
(ii) Φ * Solution: No. The empty set cannot be a universal set for non-empty sets.
(iii) {0,1,2,3,4,5,6,7,8,9,10} * Solution: Yes. All elements of A, B, and C are present in this set.
(iv) {1,2,3,4,5,6,7,8} * Solution: No. Element ‘0’ from set C is not present in this set.
Therefore, only (iii) {0,1,2,3,4,5,6,7,8,9,10} can be considered as a universal set for A, B, and C.
This provides a comprehensive set of solutions for Exercise 1.3 of RBSE Class 11 Maths Chapter 1 (Sets). Remember to understand the underlying definitions and properties of sets to solve these problems effectively.