Sets: CBSE Class 11 Maths NCERT Guide

What are Sets? Definition and Representation

In mathematics, a set is a collection of well-defined and distinct objects. Let's break this down:

• Well-defined: This means there should be no ambiguity in deciding whether an object belongs to the collection or not. For example, 'the set of vowels in the English alphabet' is well-defined because we know it's {a, e, i, o, u}. However, 'the set of intelligent students' is not well-defined because the criteria for 'intelligent' are subjective.
• Distinct: Each object in a set must be unique. We do not list the same object more than once.

There are two primary ways to represent a set:

1. Roster or Tabular Form: In this method, we list all the elements of the set, separated by commas and enclosed within curly braces {}. For instance, the set of even numbers less than 10 is written as {2, 4, 6, 8}. The order in which the elements are listed is not important.
2. Set-Builder Form: In this method, we write a variable (like x) representing each element of the set, followed by a colon or vertical bar, and then state the common property P(x) that all the elements share. It is written as {x : P(x)}. For example, the set {2, 4, 6, 8} can be written in set-builder form as {x : x is an even natural number and x < 10}.

Key Definitions: Types of Sets

Empty Set (or Null Set)

A set containing no elements at all. It is denoted by the symbol ∅ or {}. For example, the set of integers between 1 and 2.

Singleton Set

A set that has only one element. For example, {0} is a singleton set.

Finite and Infinite Sets

A set is finite if it is empty or consists of a definite number of elements. Otherwise, the set is called infinite. For example, {1, 2, 3} is finite, but the set of all natural numbers {1, 2, 3, ...} is infinite.

Equal Sets

Two sets A and B are said to be equal if they have exactly the same elements. We write A = B. For example, if A = {1, 2, 3} and B = {3, 1, 2}, then A = B.

Subset (⊂)

A set A is a subset of a set B if every element of A is also an element of B. We write A ⊂ B. The empty set ∅ is a subset of every set, and every set is a subset of itself.

Power Set P(A)

The collection of all possible subsets of a set A is called the power set of A, denoted by P(A). If a set A has n elements, then its power set P(A) will have 2^n elements.

Worked Examples: Roster Form, Set-Builder Form, and Power Sets

• Example 1: Convert from Set-Builder Form to Roster Form Write the set A = {x : x is an integer and -2 < x ≤ 4} in roster form. Step 1: Identify the condition. The condition is that x must be an integer. It must also be greater than -2 and less than or equal to 4. Step 2: List the integers that satisfy the condition. The integers greater than -2 are -1, 0, 1, 2, 3, 4, ... The integers less than or equal to 4 are ..., 2, 3, 4. We need the integers that satisfy both conditions. Step 3: Combine the conditions and write the set. The integers are -1, 0, 1, 2, 3, and 4. Final Answer: A = {-1, 0, 1, 2, 3, 4}.
• Example 2: Convert from Roster Form to Set-Builder Form Write the set B = {2, 4, 8, 16, 32} in set-builder form. Step 1: Analyze the elements to find a common property. The elements are 2, 4, 8, 16, 32. We can see that 2 = 2¹, 4 = 2², 8 = 2³, 16 = 2⁴, and 32 = 2⁵. Step 2: Describe the property using a variable. Each element is a power of 2. We can write this as 2ⁿ, where 'n' is a natural number. Step 3: Define the range of the variable. The exponent 'n' takes values from 1 to 5. So, we can write n ∈ N and 1 ≤ n ≤ 5. Final Answer: B = {x : x = 2ⁿ, where n is a natural number and 1 ≤ n ≤ 5}.
• Example 3: Find the Power Set Find the power set of the set C = {a, b}. Step 1: Recall the definition of a power set. The power set P(C) is the set of all subsets of C. Step 2: List all possible subsets of C. - The empty set is always a subset: ∅ - Subsets with one element: {a}, {b} - The set itself is always a subset: {a, b} Step 3: Write these subsets as elements of the power set. The power set is the set containing all the subsets we listed. Final Answer: P(C) = {∅, {a}, {b}, {a, b}}. (Note: Set C has 2 elements, so its power set has 2² = 4 elements, which matches our result.)

Exam Trap: Belongs To (∈) vs. Is a Subset Of (⊂)

A very common point of confusion for students is the difference between the symbols ∈ (belongs to) and ⊂ (is a subset of). Mastering this difference is key to avoiding silly mistakes in exams.

• ∈ (Belongs to): This symbol connects an element to a set. You use it to say that an item is inside a set.
• Example: If A = {5, 6, 7}, then 5 ∈ A is TRUE. 8 ∈ A is FALSE.

• ⊂ (Is a subset of): This symbol connects a set to another set. You use it to say that all elements of the first set are also contained within the second set.
• Example: If A = {5, 6, 7}, then {5, 6} ⊂ A is TRUE.

The Classic Trap:
Consider the set B = {1, 2, {3, 4}}.

• Is 3 ∈ B? False. The number 3 is not an element of B. The elements of B are 1, 2, and the set {3, 4}.
• Is {3, 4} ∈ B? True. The set {3, 4} is listed as a single element inside B.
• Is {3, 4} ⊂ B? False. For this to be true, the elements of {3, 4} (which are 3 and 4) must be elements of B. But they are not.
• Is {{3, 4}} ⊂ B? True. This is the correct way to show the subset relationship. The element is {3, 4}, so the set containing that element, which is {{3, 4}}, is a subset of B.

Practice Questions with Solutions

• Q: Which of the following are sets? Justify your answer. (i) The collection of all months of a year beginning with the letter J. (ii) The collection of the 10 most talented writers of India. A: Step 1: Analyze collection (i). The condition is 'months of a year beginning with J'. We can definitively list these: January, June, July. Since the collection is well-defined, it is a set. Step 2: Analyze collection (ii). The condition is 'most talented writers'. The term 'most talented' is subjective and varies from person to person. Since the collection is not well-defined, it is not a set. Final answer: (i) is a set. (ii) is not a set.
• Q: Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces: (i) 5 ___ A (ii) 8 ___ A (iii) 0 ___ A A: Step 1: The symbol ∈ means 'is an element of' and ∉ means 'is not an element of'. We check if the given number is present in the list of elements of set A. Step 2: For (i), the number 5 is present in set A. So, 5 ∈ A. Step 3: For (ii), the number 8 is not present in set A. So, 8 ∉ A. Step 4: For (iii), the number 0 is not present in set A. So, 0 ∉ A. Final answer: (i) ∈, (ii) ∉, (iii) ∉.
• Q: Write the set A = {1, 4, 9, 16, 25, ...} in set-builder form. A: Step 1: Observe the pattern in the elements. 1 = 1², 4 = 2², 9 = 3², 16 = 4², 25 = 5². The elements are the squares of natural numbers. Step 2: Define this property using a variable. Let the variable be x. Each x is the square of a natural number, n. So, x = n². Step 3: Specify the domain for n. The pattern continues indefinitely, so n can be any natural number (1, 2, 3, ...). We can write this as n ∈ N. Final answer: A = {x : x = n², where n ∈ N}.
• Q: Let A = {1, {2, 3}}. Are the following statements true or false? Give reasons. (i) {2, 3} ⊂ A (ii) 1 ∈ A A: Step 1: Analyze statement (i). For {2, 3} to be a subset of A, every element of {2, 3} must be an element of A. The elements of {2, 3} are 2 and 3. Step 2: Check if 2 and 3 are elements of A. The elements of A are 1 and the set {2, 3}. Neither 2 nor 3 is an element of A. Therefore, the statement is false. Step 3: Analyze statement (ii). The symbol ∈ asks if 1 is an element of set A. Step 4: Look at the elements of A = {1, {2, 3}}. The number 1 is listed as an element. Therefore, the statement is true. Final answer: (i) False, because 2 ∉ A and 3 ∉ A. (ii) True, because 1 is an element of A.

Frequently Asked Questions