Basic Concepts of Sets

Basic Concepts of Sets

• A set is a collection of distinct objects called elements or members.
• Example of a set: S = {spring, summer, fall, winter} representing seasons.
• The order of elements in a set does not matter; S = {winter, spring, fall, summer} is equivalent to the first example.

Methods to Define Sets

• Word Description: Sets can be described using natural language.
• Roster Method: Lists elements within braces, separating them with commas, such as {1, 2, 3, 4} for natural numbers less than 5.

Types of Numbers

• Natural Numbers: Also called counting numbers; denoted as N.
• Integers: Include positive, negative whole numbers, and zero.
• Rational Numbers: Can be expressed as p/q where p and q are integers, and q ≠ 0. Terminating or repeating decimals are rational numbers.
• Irrational Numbers: Cannot be expressed as a fraction, represented by non-terminating, non-repeating decimals, e.g., 0.353353335....

Set Definitions

• A well-defined set allows for clear determination of whether an item is included.
• Example of a well-defined set: letters of the English alphabet. Conversely, the set of "great songs" lacks clear criteria.
• Symbol ∈ denotes membership, while ∉ indicates non-membership.

Empty Set

• The empty set contains no elements, represented as ∅ or { }.

Set-Builder Notation

• Describes sets based on properties of their elements, e.g., {x | x ∈ N and x > 7} for natural numbers greater than 7.

Finite Sets

• A set is finite if it has a limited number of elements.
• The cardinality of a set denotes the number of elements, symbolized as n(A).
• Example: For A = {1, 4, 6, 9}, the cardinality is n(A) = 4.

Equal and Equivalent Sets

• Two sets are considered equal if they contain the same elements, regardless of order.
• Sets are equivalent if they contain the same number of elements but may differ in their actual elements.