CBSE Class 11 Maths Chapter 1 Sets: Revision Notes

Introduction to Sets and Their Representation

A set in mathematics is a well-defined collection of distinct objects. The term 'well-defined' means that it must be absolutely clear whether a particular object belongs to the set or not. For example, 'the collection of all good hockey players' is not a set because 'good' is subjective and not well-defined. However, 'the collection of all even numbers less than 10' is a set, as its members are clearly {2, 4, 6, 8}.

Objects belonging to a set are called its elements or members. Sets are typically denoted by capital letters (e.g., A, B, C), and their elements are denoted by lowercase letters (e.g., a, b, c).

There are two primary methods to represent a set:

1. Roster or Tabular Form: In this method, all elements of the set are listed, separated by commas, and enclosed within curly braces {}. The order in which the elements are listed does not matter, and elements are not generally repeated. For instance, the set of vowels in the English alphabet can be written as V = {a, e, i, o, u} or V = {e, a, i, o, u}.

1. Set-Builder Form: In this method, we describe the elements of the set by stating a common property that all elements of the set possess, and no other element outside the set possesses. It is usually written as {x : P(x)} or {x | P(x)}, where x represents an arbitrary element of the set and P(x) is the property that x must satisfy. For example, the set of all even natural numbers can be written as A = {x : x is an even natural number} or {x | x ∈ N and x is even}. This form is particularly useful for representing infinite sets or sets with complex properties where listing all elements is impractical or impossible. Understanding these representation methods is fundamental for manipulating and interpreting sets effectively in problem-solving.

Key Definitions in Set Theory

Set

A well-defined collection of distinct objects.

Element (Member)

An object belonging to a set.

Empty Set (Null Set)

A set containing no elements, denoted by Ø or {}. Its cardinality is 0.

Finite Set

A set that contains a finite or countable number of elements.

Infinite Set

A set that is not finite; it contains an uncountable number of elements.

Subset

Set A is a subset of set B (denoted A ⊆ B) if every element of A is also an element of B. Every set is a subset of itself, and the empty set is a subset of every set.

Proper Subset

Set A is a proper subset of set B (denoted A ⊂ B) if A ⊆ B and A ≠ B. This means B contains at least one element not in A.

Universal Set

A basic set that contains all possible elements under consideration for a particular context, denoted by U.

Power Set

The set of all possible subsets of a given set A, denoted by P(A). If A has n elements, P(A) has 2^n elements.

Equal Sets

Two sets A and B are equal (denoted A = B) if they have exactly the same elements. This implies A ⊆ B and B ⊆ A.

Must Remember: Key Properties & Formulas

• Order of elements does not matter in a set. {1, 2, 3} is the same as {3, 1, 2}.
• Repetition of elements is ignored in a set. {1, 1, 2, 3} is the same as {1, 2, 3}.
• The number of subsets of a set with n elements is 2^n.
• The number of proper subsets of a set with n elements is 2^n - 1.
• Cardinality of Union: n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
• Cardinality for Disjoint Sets: If A and B are disjoint (A ∩ B = Ø), then n(A ∪ B) = n(A) + n(B).
• De Morgan's Laws: 1. (A ∪ B)' = A' ∩ B' 2. (A ∩ B)' = A' ∪ B'
• Complement Laws: A ∪ A' = U (Universal Set) and A ∩ A' = Ø (Empty Set).
• Idempotent Laws: A ∪ A = A and A ∩ A = A.
• Double Complement Law: (A')' = A.

Comparison of Set Operations

Worked Examples: Applying Set Concepts

• Example 1: Set Representation Represent the set of all integers whose square is less than 10 in both Roster and Set-Builder forms. Solution: Roster Form: The integers whose squares are less than 10 are -3 (9), -2 (4), -1 (1), 0 (0), 1 (1), 2 (4), 3 (9). So, the set is A = {-3, -2, -1, 0, 1, 2, 3}. Set-Builder Form: A = {x : x ∈ Z and x² < 10}.
• Example 2: Set Operations and Cardinality Let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7}. Find A ∪ B, A ∩ B, A - B, and n(A ∪ B). Solution: A ∪ B = {1, 2, 3, 4, 5, 6, 7} A ∩ B = {4, 5} A - B = {1, 2, 3} n(A) = 5, n(B) = 4, n(A ∩ B) = 2. Using the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B) = 5 + 4 - 2 = 7. This matches the count of elements in A ∪ B.

Exam Tip: Mastering Set Theory Questions

When solving problems involving set operations or real-world scenarios, always try to draw a Venn diagram. It visually simplifies complex relationships between sets and helps you identify the correct regions corresponding to unions, intersections, differences, or complements. Pay close attention to keywords like 'exactly', 'at least one', 'none', which dictate whether you should include or exclude certain regions in your calculations. For questions on properties like De Morgan's Laws, ensure you can state and apply them correctly, perhaps by proving them with an example or using Venn diagrams. Remember that n(A) represents the number of elements in set A, not the set itself. A common trap is confusing {Ø} (a set containing the empty set) with Ø (the empty set itself).

Practice Questions with Solutions

• Q: If A = {x : x is a prime number less than 10}, write A in roster form and find n(A). A: A = {2, 3, 5, 7}; n(A) = 4.
• Q: Given U = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, what is A' (complement of A)? A: A' = {2, 4, 6}.
• Q: Is the set {x : x ∈ R, x² = -1} an empty set or a non-empty set? Explain. A: It is an empty set (Ø). There is no real number (x ∈ R) whose square is -1.
• Q: If a set has 4 elements, how many subsets does it have, and how many proper subsets? A: Number of subsets = 2^4 = 16. Number of proper subsets = 2^4 - 1 = 15.

Frequently Asked Questions