1.1 Definition and notation of sets
2 min read•august 7, 2024
Sets are the building blocks of mathematics, forming collections of distinct objects. This section introduces fundamental concepts like elements, empty sets, and finite vs. infinite sets, laying the groundwork for understanding theory.
We explore different ways to represent sets, including and . These tools help us clearly communicate set contents and properties, essential for working with sets in various mathematical contexts.
Set Definitions
Fundamental Concepts
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- Set represents a well-defined collection of distinct objects
- refers to each individual object within a set
- contains no elements and is denoted by the symbol ∅ or {}
- has a specific number of elements that can be counted (integers from 1 to 10)
- has an endless number of elements that cannot be counted (real numbers between 0 and 1)
Types of Sets
- Sets can be classified based on the number of elements they contain
- Finite sets have a countable number of elements and can be explicitly listed (days of the week)
- Infinite sets have an uncountable number of elements and cannot be explicitly listed (points on a line segment)
- Empty set is a unique set that contains no elements and is a of every set
- contain at least one element and can be either finite or infinite
Set Notation
Roster Notation
- Roster notation is a method of representing a set by listing its elements within
- Elements are separated by commas and enclosed in curly braces { }
- Roster notation is useful for representing small, finite sets (vowels in the English alphabet {a, e, i, o, u})
- Order of elements in roster notation does not matter as sets are unordered collections
- Repeated elements are only listed once in roster notation as sets contain distinct elements
Ellipsis and Set-Builder Notation
- (...) is used in roster notation to represent a pattern of elements that continues indefinitely
- Ellipsis is commonly used for representing infinite sets or sets with a clear pattern (even numbers {2, 4, 6, ...})
- Set-builder notation is used to describe a set based on a property that its elements satisfy
- Set-builder notation uses a vertical bar | to separate the element symbol from the property ({x | x is a positive integer less than 10})
- Curly braces are used in both roster and set-builder notation to denote the boundaries of the set
Key Terms to Review (11)
Curly Braces: Curly braces, represented as `{` and `}`, are symbols used in set notation to define a set by enclosing its elements. This notation is essential for clearly distinguishing the elements of a set from other mathematical expressions, making it easier to understand the boundaries of the set being defined. The use of curly braces is a foundational concept in set theory, allowing for precise communication about collections of objects.
Element: An element is an individual object or member that belongs to a set. In set theory, elements are the basic building blocks of sets, and understanding their role helps in grasping the broader concepts of set membership and relationships between sets.
Ellipsis: In set theory, an ellipsis is a notation used to indicate that a pattern continues indefinitely or that there are additional elements in a set that follow a specified rule. It allows for the concise representation of sets without listing every single element, which can be especially useful when dealing with large or infinite sets. This notation helps to simplify the expression of sets and makes it easier to understand the underlying relationships between elements.
Empty set: The empty set is a unique set that contains no elements, represented by the symbols ∅ or {}. It serves as a fundamental concept in set theory, highlighting the idea that a set can exist without containing any objects, and connects to various principles like membership and operations involving sets.
Finite set: A finite set is a collection of distinct objects that has a limited number of elements. This means you can count the members of the set and arrive at a specific integer, unlike infinite sets which do not have a definite size. The concept of finite sets plays a crucial role in understanding various set operations, properties, and foundational theories in mathematics.
Infinite set: An infinite set is a collection of elements that has no end or limit; it can be counted indefinitely. This concept is important because it distinguishes between finite sets, which have a specific number of elements, and infinite sets, which can be either countably infinite, like the set of natural numbers, or uncountably infinite, like the set of real numbers. Understanding infinite sets helps to clarify various foundational aspects of mathematics and set theory.
Non-empty Sets: A non-empty set is a collection of distinct elements that contains at least one member. This concept is foundational in set theory as it distinguishes sets that have elements from those that do not, known as empty sets. Understanding non-empty sets helps clarify operations and relations in set theory, since they serve as the basis for constructing other sets and performing various mathematical operations.
Roster Notation: Roster notation is a way of specifying a set by listing its elements within curly braces. Each element is separated by a comma, and the notation gives a clear and concise way to represent the members of the set without ambiguity. This method is particularly useful when dealing with finite sets, allowing for easy identification of all contained elements.
Set: A set is a well-defined collection of distinct objects, considered as an object in its own right. Sets are fundamental in mathematics and serve as the building blocks for various concepts and structures, allowing for the organization and analysis of elements in different mathematical contexts.
Set-builder notation: Set-builder notation is a mathematical shorthand used to describe sets by specifying the properties that its members must satisfy. This notation allows for a clear and concise way to define sets, particularly those that are infinite or defined by a particular rule. It connects closely with the basic definitions and notation of sets, providing a more sophisticated framework for representing collections of elements, and also plays a critical role in discussing foundational issues in set theory, such as Russell's Paradox.
Subset: A subset is a set where every element of that set is also contained within another set. Understanding subsets is crucial because they form the basis for defining relationships between sets, including set membership, unions, intersections, and various operations performed on sets.