Applied Mathematics: Operations on Sets and Venn Diagrams

Introduction to Set Theory

• Definition of a Set: A set is a collection of well-defined objects which are distinct from each other.
• Elements: The objects contained within a set are referred to as its elements.
• Notation Conventions:
  • Sets are typically denoted by capital letters, such as $A, B, C, \dots$
  • Elements of sets are typically denoted by small letters, such as $a, b, c, \dots$
• Illustrative Example: The set of all even natural numbers less than 10 can be represented as:
  • $N = \{2, 4, 6, 8\}$

Methods for Describing a Set

• Roster Method:
  • In this method, a set is described by listing all its elements, separated by commas, within curly braces ($\{\}$).
  • Alternative Names: This method is also known as the listing method or the tabular form method.
  • Example: $A = \{a, e, i, o, u\}$
• Set-Builder Method:
  • In this method, we do not list elements but instead write down a rule or property that defines all elements of the set.
  • Example: $A = \{x : x \text{ is a vowel of English alphabet}\}$

Classification and Types of Sets

• Finite Set: A set containing a finite number of elements or no elements at all is called a finite set.
  • Example: The set of all persons in India.
• Infinite Set: A set containing an infinite number of elements is categorized as an infinite set.
• Empty Set (or Null Set):
  • A set containing no elements is called an empty set.
  • Notation: It is represented by the symbol $\{\}$ or $\phi$ (read as phi).
  • Example: The set of all whole numbers less than $0$.
• Singleton Set:
  • A set that contains exactly one single element.
  • Example: The set of all even prime numbers (In this case, the set would be $\{2\}$).

Cardinality of Sets

• Definition: The total number of unique elements present in a given finite set is called the cardinal number of that set.
• Mathematical Notation: For a set $A$, the cardinal number is denoted by $n(A)$.
• Application Example:
  • Given a set $P = \{5, 15, 25, 35, 45\}$
  • The cardinal number is $n(P) = 5$

Set Relationships: Equality and Disjointness

• Equal Sets:
  • Two sets $A$ and $B$ are considered equal if every element of $A$ is a member of $B$ and every element of $B$ is a member of $A$.
  • Notation: $A = B$
  • Example: Set $A = \{2, 4, 6, 8, 10\}$ compared against set $B = \{\text{all the even natural numbers less than or equal to } 10\}$. These two sets are equal.
• Disjoint Sets:
  • Two sets $A$ and $B$ are called disjoint if they have no elements in common.
  • Visual/Symbolic Logic: If $A$ and $B$ have no intersection, then $A \cap B = \phi$.
  • Example:
    • $A = \{2, 4, 6, 8, 10, 12\}$
    • $B = \{1, 3, 5, 7, 9, 11\}$
    • Therefore, $A$ and $B$ are disjoint sets.

Subsets and Supersets

• Subset Definition: For any two sets $A$ and $B$, if every element of $A$ is also an element of $B$, then $A$ is called the subset of $B$.
• Superset Definition: Conversely, if $A$ is a subset of $B$, then $B$ is called the superset of $A$.
• Notation:
  • Subset: $A \subset B$ (The symbol '$\subset$' stands for 'is a subset of').
  • Superset: $B \supset A$
• Core Principles of Subsets:
  • The Empty Set: The empty set ($\phi$) is a subset of every set.
  • Self-Inclusion: Every set is a subset of itself (i.e., $A \subset A$ and $B \subset B$).
  • Quantifying Subsets: The total number of possible subsets of a set $A$ having $n$ elements is calculated by the formula $2^n$.
• Example: The set of all prime numbers is a subset of the set of all whole numbers.

Worked Example: Finding Cardinality

• Problem: Find the cardinal number of a set $A$ of the composite numbers between $10$ and $25$.
  • (A) $4$
  • (B) $6$
  • (C) $8$
  • (D) $9$
  • (E) None of these
• Explanation and Solution:
  • First, list the elements of set $A$ (composite numbers between $10$ and $25$):
  • $A = \{12, 14, 15, 16, 18, 20, 21, 22, 24\}$
  • Count the elements: there are $9$ items in the set.
  • Therefore, the cardinal number $n(A) = 9$.
  • Answer: (D)