Sets Class 11 Maths Chapter Notes

Key Definitions in Sets

Set

A well-defined collection of distinct objects. The objects in a set are called its elements or members.

Roster or Tabular Form

A method of representing a set where all the elements are listed, separated by commas, and enclosed within braces { }.

Set-builder Form

A method of representing a set by stating a common property that all its elements possess. Syntax: {x : P(x)}, where P(x) is the property.

Empty Set (∅ or { })

A set containing no elements. It is also called the null set or void set.

Subset (⊆)

A set A is a subset of set B if every element of A is also an element of B. Notation: A ⊆ B.

Proper Subset (⊂)

A set A is a proper subset of set B if A is a subset of B and A ≠ B. This means B contains at least one element not in A.

Power Set (P(A))

The collection of all subsets of a set A. If a set A has 'n' elements, its power set P(A) has 2ⁿ elements.

Universal Set (U)

A basic set that contains all the elements under consideration in a particular context. All other sets are subsets of the universal set.

Cardinal Number (n(A))

The number of distinct elements in a finite set A.

Types of Sets: A Quick Comparison

Operations on Sets

Understanding operations on sets is fundamental to solving problems. These operations combine or modify sets to create new ones. They are visually represented using Venn diagrams.

1. Union of Sets (A ∪ B): The union of two sets A and B is the set containing all elements that are in A, or in B, or in both. In set-builder form, A ∪ B = {x : x ∈ A or x ∈ B}. Think of it as 'merging' the two sets and removing duplicates. For example, if A = {1, 2} and B = {2, 3}, then A ∪ B = {1, 2, 3}.

1. Intersection of Sets (A ∩ B): The intersection of two sets A and B is the set of all elements that are common to both A and B. In set-builder form, A ∩ B = {x : x ∈ A and x ∈ B}. This operation finds the 'overlap' between the sets. For the same sets A and B, A ∩ B = {2}.

1. Difference of Sets (A – B): The difference of set A and set B is the set of elements which are in A but not in B. In set-builder form, A – B = {x : x ∈ A and x ∉ B}. It's important to note that A – B is not the same as B – A. Using our example, A – B = {1} and B – A = {3}.

1. Complement of a Set (A'): The complement of a set A, with respect to a universal set U, is the set of all elements in U that are not in A. It is denoted by A' or Aᶜ. In set-builder form, A' = {x : x ∈ U and x ∉ A}. Essentially, A' = U – A. If U = {1, 2, 3, 4, 5} and A = {1, 2}, then A' = {3, 4, 5}.

Must-Remember Formulas and Properties

• Number of subsets: For a set with 'n' elements, the total number of subsets is 2ⁿ.
• Number of proper subsets: For a set with 'n' elements, the number of proper subsets is 2ⁿ - 1.
• Cardinality Formula (Union): For any two finite sets A and B, n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
• Cardinality Formula (Three Sets): n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C).
• De Morgan’s Laws: Crucial for proofs and simplification. (i) (A ∪ B)' = A' ∩ B' (ii) (A ∩ B)' = A' ∪ B'
• Complement Law: A ∪ A' = U and A ∩ A' = ∅.
• Idempotent Laws: A ∪ A = A and A ∩ A = A.
• Identity Laws: A ∪ ∅ = A and A ∩ U = A.
• Commutative Laws: A ∪ B = B ∪ A and A ∩ B = B ∩ A.
• Relationship between Difference and Intersection: A – B = A ∩ B'.

Worked Examples

• If A = {x : x is a natural number and x < 5} and B = {x : x is an even prime number}, find A ∪ B and A ∩ B. Step 1: Write sets in Roster Form. A = {1, 2, 3, 4} B = {2} (2 is the only even prime number) Step 2: Find the Union. A ∪ B = {1, 2, 3, 4} ∪ {2} = {1, 2, 3, 4} Step 3: Find the Intersection. A ∩ B = {1, 2, 3, 4} ∩ {2} = {2}
• In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play both cricket and football? Step 1: Identify the given values. Let C be the set of students who like cricket and F be the set of students who like football. Total students, n(C ∪ F) = 35 (since each student likes at least one game). n(C) = 24 n(F) = 16 Step 2: Use the cardinality formula. n(C ∪ F) = n(C) + n(F) – n(C ∩ F) 35 = 24 + 16 – n(C ∩ F) 35 = 40 – n(C ∩ F) Step 3: Solve for the intersection. n(C ∩ F) = 40 – 35 = 5. Therefore, 5 students like to play both games.

Exam Traps and Scoring Tips

1. Notation is Key: Be very precise with symbols. ⊂ (proper subset) and ⊆ (subset) are different. ∈ (belongs to) is used for elements, while ⊆ (is a subset of) is used for sets. For example, 2 ∈ {1, 2, 3} but {2} ⊆ {1, 2, 3}. Using the wrong symbol can cost you marks.

2. Power Set Confusion: Remember that the power set P(A) is a set of sets. Its elements are the subsets of A. The empty set ∅ and the set A itself are always elements of P(A).

3. Word Problems: For word problems involving 'at least one', 'both', or 'only one', always draw a Venn diagram. It helps visualize the problem and ensures you use the correct formula. Label each region of the diagram clearly before calculating.

4. De Morgan's Law Application: When asked to prove something or simplify a complex set expression, always check if De Morgan's laws can be applied. They are frequently tested.

Quick Revision Check

• If set A has 4 elements, how many elements are in its power set, P(A)? The number of elements in the power set is 2ⁿ. So, P(A) will have 2⁴ = 16 elements.
• Write the set A = {x | x is an integer and -3 < x < 3} in roster form. A = {-2, -1, 0, 1, 2}. Note that -3 and 3 are not included because of the strict inequality (<).
• What is the difference between the empty set (∅) and the set {∅}? The empty set (∅) is a set with no elements. The set {∅} is a singleton set whose only element is the empty set itself. n(∅) = 0, but n({∅}) = 1.
• Given A = {a, b} and B = {c, d}, are they equal or equivalent? They are equivalent because they have the same number of elements (n(A) = 2, n(B) = 2). They are not equal because they do not have the same elements.

Frequently Asked Questions about Sets

Frequently Asked Questions