Introduction to Sets and Type of Sets

Data Science and A.I. Lecture  Series

Introduction

• A set is a well-defined collection of distinct objects.
• Examples of collections:
  • Books in a library.
  • Natural numbers that are factors of a given number.
  • States in a country.
• Sets are fundamental in mathematics and are used in many areas, including geometry, sequences, and probability.
• Understanding sets is essential for grasping advanced mathematical concepts.

Sets

Definition of Sets

• A set is generally denoted by capital letters (e.g., \(A, B, C\)).
• Elements of a set are called members, denoted by small letters (e.g., \(a, b, c\)).
• If \(a\) is an element of set \(A\), we write \(a \in A\).
• If \(a\) is not an element of \(A\), we write \(a \notin A\).

Examples of Sets

• Well-defined collections that form sets:
  • Natural numbers less than 5: \(\{1, 2, 3, 4\}\).
  • Letters in the word “ASSIGNMENT”: \(\{A, S, I, G, N, M, E, T\}\).
• Collections that do not form sets:
  • “Good cricketers” (subjective).
  • “Honest students” (subjective).

Methods of Representing a Set

Roster Method

• Lists all elements explicitly within curly brackets.
• Examples:
  • \(A = \{a, e, i, o, u\}\) (vowels of English alphabets).
  • \(N = \{1, 2, 3, 4, 5, \dots\}\) (natural numbers).
  • \(W = \{0, 1, 2, 3, 4, 5, \dots\}\) (whole numbers).
  • \(Z = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}\) (integers).
  • \(E = \{2, 4, 6, 8, 10, 12, \dots\}\) (even natural numbers).
  • \(O = \{1, 3, 5, \dots\}\) (odd natural numbers).
  • \(P = \{2, 3, 5, 7, 11, 13, 17, \dots\}\) (prime numbers).

Set-Builder Method

• Describes properties of elements in the set.
• Examples:
  • \(A = \{x : x \text{ is a vowel of English alphabet}\}\).
  • \(A = \{x : x \text{ is a natural number and } x \text{ is a multiple of 3}\}\).
  • \(A = \{x : x \text{ is a factor of 10 and } x > 0\}\).
  • \(Q = \{x : x = \frac{p}{q}, p, q \in \mathbb{Z}, q \neq 0\}\) (rational numbers).

Types of Sets

Null Set (Empty Set)

• A set with no elements.
• Denoted by \(\emptyset\) or \(\{\}\).
• Example: \(A = \{x : x \text{ is a natural number, } 1 < x < 2\}\).

Singleton Set

• A set with exactly one element.
• Example: \(A = \{x : x \text{ is an even prime number}\} = \{2\}\).

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