Sets Chapter Notes | Mathematics Class 8 ICSE PDF Download

Introduction

Set

• A set is a collection of clearly defined objects, such as numbers, names, or items.
• Objects must be specific for the collection to qualify as a set.
  • Example: The collection of students with heights between 135 cm and 160 cm is a set because it is well-defined.
  • Counter example: The collection of "tall students" is not a set because "tall" is subjective and not clearly defined.

Elements

• Elements or members are the individual objects that make up a set.
• Sets are usually denoted by capital letters (e.g., A, B, V).
• Elements are written inside curly braces { } and separated by commas.
• Examples:
  • Set A = {John, Geeta, Amit, Rohit} represents a set of names.
  • Set V = {a, e, i, o, u} represents the vowels of the English alphabet.
• The symbol ∈ means "belongs to," e.g., a ∈ V means "a" is an element of set V.
• The symbol ∉ means "does not belong to," e.g., b ∉ V means "b" is not an element of set V.

Representation of a Set

Sets can be represented in two main ways: Roster Form and Set-Builder Form.

Roster (or Tabular) Form

• In Roster Form, all elements of the set are listed inside curly braces { }, separated by commas.
• Examples:
  • Set A = {2, 5, 7, 9, 15} lists the numbers directly.
  • Set of integers Z = {…, -2, -1, 0, 1, 2, 3, …}.
  • Set of whole numbers W = {0, 1, 2, 3, 4, …}.
  • Set of natural numbers N = {1, 2, 3, 4, …}.
• The order of elements does not matter, e.g., {a, b, c} is the same as {b, a, c}.
• Each element is written only once, e.g., {2, 3, 3, 2, 4} is simplified to {2, 3, 4}.
  • Example: The set of letters in "ALLAHABAD" is {a, l, h, b, d}, with no duplicates.

Set-Builder Form (Rule Method)

• In Set-Builder Form, a rule or formula describes the elements of the set without listing them.
• Format: {x : condition on x}, read as "the set of all x such that the condition holds."
• Examples:
  • A = {x : x ∈ N and x < 7} describes natural numbers less than 7, which is {1, 2, 3, 4, 5, 6}.
  • Set A = {2, 3, 4, 5} can be written as {x : x ∈ N, 2 ≤ x < 6} or {x : x ∈ N, 1 < x ≤ 5}.
  • Set C = {1, 3, 5, 7, 9, 11} can be written as {x : x = 2n-1, n ∈ N, n ≤ 6} or {x : x = 2n+1, n ∈ W, n ≤ 5}.
  • Set D = {2, 4, 6, 8} can be written as {x : x = 2n, n ∈ N, n ≤ 4}.

Cardinal Number of a Set

• The cardinal number of a set is the number of elements it contains, denoted by n(A).
• Examples:
  • For A = {c, d, f}, n(A) = 3.
  • For B = {2, 4, 5, 6}, n(B) = 4.

Types of Sets

Sets can be classified based on the number and nature of their elements.

Finite Set

• A set with a limited number of elements is called a finite set.
• Examples:
  • The set of boys in a class.
  • {x : x is a member of a particular family}.
  • {3, 4, 5, …, 100} has 98 elements.

Infinite Set

• A set with an unlimited number of elements is called an infinite set.
• Examples: 
  • {x : x is a living thing}.
  • {x : x ∈ W and x > 1000}.

Singleton or Unit Set

• A set with exactly one element is called a singleton or unit set.
• Examples: 
  • {x : x is President of India}.
  • The set of whole numbers between 6 and 8 is {7}.
  • {x : 2x - 1 = 3} is {2}.

Empty or Null Set

• A set with no elements is called an empty or null set, denoted by Ø (pronounced "oe").
• Examples:
  • The set of odd numbers between 7 and 9 is Ø.
  • {x : x ∈ N and x < 1} is Ø.
• The cardinal number of the empty set is 0, i.e., n(Ø) = 0.

Note: 

• {0} is not empty because it contains the element 0.
• {Ø} is not empty because it contains the empty set as an element.

Joint or Overlapping Sets

• Two sets are joint or overlapping if they have at least one element in common.
• Example: A = {5, 7, 9, 11} and B = {6, 9, 12, 15} are overlapping because 9 is common.

Disjoint Sets

• Two sets are disjoint if they have no elements in common.
• Examples: 
  • A = {5, 7, 9, 11} and B = {4, 6, 8, 10} are disjoint.
  • A = {students of Sophia Girls' School} and B = {students of St. Mary's Academy} are disjoint.

Equivalent Sets

• Two sets are equivalent if they have the same number of elements, denoted by A ↔ B.
• Example: A = {a, b, c} and B = {x, y, z} are equivalent because n(A) = n(B) = 3.

Equal Sets

• Two sets are equal if they contain exactly the same elements.
• Example: A = {1, 2, 3, 4, 5} and B = {x : x ∈ N and x < 6} are equal because they have identical elements.

Note: Equal sets are always equivalent, but equivalent sets are not necessarily equal.

Subset

• A set A is a subset of set B if every element of A is also in B, denoted by A ⊆ B.
• A ⊆ B is read as "A is a subset of B" or "A is contained in B."
• Examples:
  • A = {5, 6, 7} and B = {2, 3, 4, 5, 6, 7}, so A ⊆ B.
  • If A = {students of a school} and B = {class VIII students of the school}, then B ⊆ A.
  • If P = {even natural numbers} and Q = {multiples of 4}, then Q ⊆ P.
• Every set is a subset of itself, e.g., A ⊆ A.
• The empty set Ø is a subset of every set, e.g., Ø ⊆ A.
• If A ⊆ B and B ⊆ A, then A = B.

Proper Subset

• A set A is a proper subset of set B if all elements of A are in B and B has at least one element not in A, denoted by A ⊂ B.
• A ⊂ B is read as "A is a proper subset of B."
• Examples: 
  • A = {5, 6, 7} and B = {3, 5, 6, 7, 8}, so A ⊂ B.
  • The set of natural numbers N is a proper subset of whole numbers W, i.e., N ⊂ W.

Number of Subsets and Proper Subsets of a Given Set

• For a set with n elements:
• The number of subsets is 2ⁿ.
• The number of proper subsets is 2^{n - 1}.
• Examples:
  • For a set with 0 elements (Ø), number of subsets = 2⁰ = 1, proper subsets = 0.
  • For {a}, number of subsets = 2¹ = 2 (Ø, {a}), proper subsets = 1 (Ø).
  • For {a, b}, number of subsets = 2² = 4 (Ø, {a}, {b}, {a, b}), proper subsets = 3 (Ø, {a}, {b}).
  • For {a, b, c}, number of subsets = 2³ = 8 (Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}), proper subsets = 7 (all except {a, b, c}).
• No set is a proper subset of itself.

Super Set

• If A is a subset of B, then B is a super set of A, denoted by B ⊇ A.
• B ⊇ A is read as "B is a super set of A."

Universal Set

• A universal set contains all sets under consideration as its subsets, denoted by ξ or U.
• Examples: 
  • For A = {5, 6, 7, 8}, B = {1, 3, 5, 7}, C = {4, 6, 8, 10}, a universal set could be {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The choice of a universal set is not unique.
  • For A = {2, 4, 6, 7}, B = {3, 5, 7, 9, 11}, C = {0, 4, 8, 12, 16}, possible universal sets include {x : x ∈ W and x ≤ 16}, {x : x ∈ W}, or {x : x ∈ Z, 0 ≤ x < 17}.
  • For D = {students of class 9}, possible universal sets include {students of the school} or {students of the town}.

Complement Set

• The complement of a set A, denoted A', is the set of all elements in the universal set ξ that are not in A.
• Examples: 
  • If ξ = {1, 2, 3, 4, 5, 6} and A = {2, 3, 5}, then A' = {1, 4, 6}.
  • If ξ = {x : x ∈ N} and A = {even natural numbers}, then A' = {odd natural numbers}.
• A set and its complement are disjoint, meaning they have no common elements.
• The complement of the empty set is the universal set, i.e., Ø' = ξ.
• The complement of the universal set is the empty set, i.e., ξ' = Ø.

Set Operations

Union of Two Sets

• The union of sets A and B, denoted A ∪ B, includes all elements that are in A, in B, or in both.
• Examples: 
  • If A = {5, 6, 7} and B = {6, 8}, then A ∪ B = {5, 6, 7, 8}.
  • If E = {numbers divisible by 2} and F = {numbers divisible by 3}, then E ∪ F = {numbers divisible by 2 or 3 or both}.
• Union is commutative: A ∪ B = B ∪ A.
• Union is associative: A ∪ (B ∪ C) = (A ∪ B) ∪ C.
• A is a subset of A ∪ B, i.e., A ⊂ (A ∪ B).
• If A ⊂ B, then A ∪ B = B.
• The union of a set and its complement is the universal set: A ∪ A' = ξ.
• The union of a set and the empty set is the set itself: A ∪ Ø = A.
• The union of a set and the universal set is the universal set: A ∪ ξ = ξ.

Intersection of Two Sets

• The intersection of sets A and B, denoted A ∩ B, includes all elements common to both A and B.
• Examples: 
  • If A = {5, 6, 7} and B = {6, 8}, then A ∩ B = {6}.
  • If E = {numbers divisible by 2} and F = {numbers divisible by 3}, then E ∩ F = {numbers divisible by 6}.
• Intersection is commutative: A ∩ B = B ∩ A.
• Intersection is associative: A ∩ (B ∩ C) = (A ∩ B) ∩ C.
• A ∩ B is a subset of A and B: (A ∩ B) ⊂ A and (A ∩ B) ⊂ B.
• If A and B are disjoint, then A ∩ B = Ø.

Difference of Two Sets

• The difference of sets A and B, denoted A - B or A \ B, is the set of elements that are in A but not in B.
• It is also written as A - B = {x : x ∈ A and x ∉ B}.
• Examples: 
  • If A = {5, 6, 7} and B = {6, 8}, then A - B = {5, 7}.
  • If A = {1, 2, 3, 4} and B = {2, 4, 6}, then A - B = {1, 3}.
• The difference is not commutative: A - B ≠ B - A.
• Example: For the above sets, B - A = {6}.
  • If A and B are disjoint, then A - B = A.
  • If A ⊆ B, then A - B = Ø.

Distributive Laws

Distributive laws describe how union and intersection operations distribute over each other.

• First Distributive Law: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
  • This means the union of A with the intersection of B and C is the same as the intersection of the unions of A with B and A with C.
• Second Distributive Law: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
  • This means the intersection of A with the union of B and C is the same as the union of the intersections of A with B and A with C.

De Morgan's Laws

De Morgan's laws relate the complement of unions and intersections of sets.

• First De Morgan's Law:(A ∪ B)' = A' ∩ B'.
  • This means the complement of the union of A and B is the intersection of their complements.
• Second De Morgan's Law:(A ∩ B)' = A' ∪ B'.
  • This means the complement of the intersection of A and B is the union of their complements.
• Example: If ξ = {1, 2, 3, 4, 5}, A = {1, 2}, B = {2, 3}, then (A ∪ B)' = {4, 5} = A' ∩ B', and (A ∩ B)' = {1, 3, 4, 5} = A' ∪ B'.

Cardinal Property

• The cardinal property helps calculate the number of elements in the union of two sets.
• Formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
• This accounts for elements counted twice in A and B by subtracting the number of common elements.
• Example: If n(A) = 20, n(B) = 25, and n(A ∩ B) = 10, then n(A ∪ B) = 20 + 25 - 10 = 35.
• For 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).

Venn Diagram

• A Venn diagram is a visual tool to represent sets and their relationships using circles within a rectangle (the universal set).
• Each circle represents a set, and the rectangle represents the universal set ξ.
• Overlapping circles indicate common elements (intersection).
• Example: For A = {5, 6, 7} and B = {6, 8}, the Venn diagram shows 6 in the overlapping region, 5 and 7 in A’s circle, and 8 in B’s circle.
• Union (A ∪ B) is represented by the entire area covered by both circles.
• Intersection (A ∩ B) is the overlapping area.
• Complement (A') is the area outside A’s circle but within the universal set.
• Difference (A - B) is the part of A’s circle not overlapping with B.

Using Venn Diagram to Show Relationship Between Sets

• Venn diagrams illustrate relationships like subsets, unions, intersections, and differences.
• For overlapping sets A and B, the diagram shows:
  • A ∩ B as the common region.
  • A ∪ B as the combined region of both circles.
  • (A ∩ B)' as the area outside the common region but within ξ.
  • (A ∪ B)' as the area outside both circles but within ξ.
  • A' ∩ B as the part of B’s circle not in A.
  • For disjoint sets, circles do not overlap, and A ∩ B = Ø.
  • For subsets, one circle (A) is completely inside another (B) if A ⊆ B.
  • A - B as the part of A’s circle not in B.
• Example: If ξ = {12, 13, 14, 15, 16, 17, 18, 19}, A = {12, 15, 18}, B = {12, 14, 15, 16}, the Venn diagram shows 12 and 15 in the overlapping region, 18 in A’s circle, and 14 and 16 in B’s circle.