Set Theory Chapter Notes | Mathematics for GCSE/IGCSE - Class 10 PDF Download

Introduction 

Imagine organising your favourite toys, books, or even friends into groups based on what makes them special—like all the red toys in one box or all your friends who love soccer in one team. Set theory is like that! It’s a fun and logical way to group things together based on their unique qualities. In this chapter, we’ll dive into the world of sets, learning how to describe, organize, and work with collections of objects, people, or ideas. From understanding what makes a set to performing cool operations like combining or comparing them, set theory helps us make sense of the world in a clear and exciting way. Let’s explore how to classify things, use special symbols, and even count the number of items in these groups!

Set

• A set is a group of clearly defined and unique objects, called elements or members.
• Clearly defined means we can easily tell if something belongs to the set or not.
• Unique means no two elements in a set are the same.
• Sets are denoted by capital letters (e.g., A, B, X, Y).
• Elements are denoted by small letters (e.g., a, b, x, y).
• The symbol ∈ means "belongs to," and ∉ means "does not belong to."

Example: The collection of positive numbers less than 11 is a set because we can clearly determine which numbers belong (e.g., 1, 2, ..., 10). However, the collection of intelligent girls in a class is not a set because "intelligent" is not clearly defined.

Notation and Representation of Sets

• Sets can be represented in two ways: roster form and set-builder form.

Roster or Tabular Form

• In roster form, all elements of a set are listed inside curly brackets { }, separated by commas.
• Elements are written without repetition.
• Example:

  The set A of the first three even natural numbers is written as A = {2, 4, 6}.

Set-Builder or Rule Form

• In set-builder form, a set is described by a common property of its elements inside curly brackets.
• It uses a variable (e.g., x) and a condition, written as {x | condition} or {x : condition}, where | or : means "such that."
• Example:

  The set Y of natural numbers less than 10 is written as Y = {x | x is a natural number less than 10}.

Types of Sets

Finite Set

• A finite set has a countable number of elements.
• Example:

  The set A = {x | x is a natural number less than 25} is a finite set because it has 24 elements (1, 2, ..., 24).

Infinite Set

• An infinite set has an uncountable number of elements.
• Example:

  The set of all stars in the sky is an infinite set because we cannot count all the stars.

Singleton Set

• A singleton set contains exactly one element.
• Example:

  The set A = {a} is a singleton set because it has only one element, a.

Empty Set

• An empty set, also called a null set, has no elements and is denoted by φ or {}.
• Example:

  The set X = {x | x < 1, x ∈ N} is an empty set because no natural number is less than 1.

Equal Sets

• Two sets are equal if they have exactly the same elements, regardless of order.
• Denoted as X = Y.
• Example:

  If X = {x | x is a letter in the word FOLLOW} and Y = {x | x is a letter in the word FOWL}, then X = {F, O, L, W} and Y = {F, O, W, L}, so X = Y.

Equivalent Sets

• Two sets are equivalent if they have the same number of elements.
• Denoted as X ↔ Y.
• Example:

  If X = {1, -1, 0} and Y = {a, b, c}, both have 3 elements, so X ↔ Y.

Disjoint Sets

• Two sets are disjoint if they have no common elements.
• Example:

  If A = {1, 2, 3, 4, 5} and B = {6, 7, 8, 9, 10}, then A and B are disjoint sets because they share no elements.

Overlapping Sets

• Two sets are overlapping if they have at least one common element.
• Example:

  If X = {x | x is a letter in the word INDIGO} and Y = {x | x is a letter in the word GREEK}, then X and Y are overlapping sets because both contain the element G.

Cardinal Number

• The number of distinct elements in a finite set is its cardinal number, denoted by n(Y).
• The cardinal number of an empty set is 0.
• The cardinal number of an infinite set is not defined.
• Example:

  If X = {Delhi, Mumbai, Chennai}, then n(X) = 3 because the set has 3 distinct elements.

Subsets

• A set X is a subset of set Y if every element of X is also in Y, denoted as X ⊆ Y.
• Y is a superset of X, denoted as Y ⊇ X.
• If X ⊆ Y and Y has at least one element not in X, then X is a proper subset of Y, denoted as X ⊂ Y.
• Every set is a subset of itself (A ⊆ A).
• The empty set is a subset of every set.
• If a set has n elements, it has 2ⁿ subsets and 2ⁿ - 1 proper subsets.

Example: The set {0, 2, 4} has 2³ = 8 subsets: φ, {0}, {2}, {4}, {0, 2}, {2, 4}, {0, 4}, {0, 2, 4}.

Universal Set

• A universal set contains all sets under consideration as its subsets, denoted by U or ξ.
• It may include extra elements not in the given sets.
• The universal set is not unique and depends on the context.
• Example:

  If A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 6}, then {1, 2, 3, 4, 5, 6, 8, 10} is a universal set for A and B.

Operations on Sets

Union of Sets

• The union of sets A and B, denoted A ∪ B, includes all elements in A, B, or both.
• Common elements are included only once.
• The union is commutative: A ∪ B = B ∪ A.
• The union is associative: A ∪ (B ∪ C) = (A ∪ B) ∪ C.
• Example:

  If A = {4, 6, 8, 10} and B = {3, 7, 9}, then A ∪ B = {4, 6, 8, 10, 3, 7, 9}.

Intersection of Sets

• The intersection of sets A and B, denoted A ∩ B, includes only the elements common to both A and B.
• Example:

  If A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12, 15}, then A ∩ B = {6, 12}.

Difference of Two Sets

• The difference A - B includes elements in A that are not in B.
• The difference is not commutative: A - B ≠ B - A.
• Steps to find A - B:
  • Identify common elements in A and B.
  • Remove these common elements from A to get A - B.
• Example:

  If A = {9, 11, 13, 17, 21} and B = {5, 7, 8, 9, 11, 15, 17}, then A - B = {13, 21}.

Cardinal Properties of Sets

• For any non-empty sets A and B:
• n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
• If A and B are disjoint (A ∩ B = φ), then n(A ∪ B) = n(A) + n(B).
• n(A - B) = n(A) - n(A ∩ B)
• n(B - A) = n(B) - n(A ∩ B)

Example: If P ∪ Q has 60 elements, P has 28 elements, and Q has 42 elements, then find n(P ∩ Q) ?

Solution: Putting the given values

n(P ∩ Q) = n(P) + n(Q) - n(P ∪ Q) = 28 + 42 - 60 = 10.