Domain and Range: Set Notation and Notation Basics (R.3) - Fri 8/15
Set Notation Overview
Topic context: This is section R.3 (Review) on domain and range, tied to MyLab Math Pearson eText and the textbook's sectioning. The instructor notes that R stands for Review and the section covers foundational notation used for sets.
Overall goal: Build comfort with how we denote sets and later how to analyze domain and range from graphs and from functions.
Quick recap of where this fits: This material is described as review, connecting to real numbers and the kinds of sets we’ll encounter in this course (real numbers, natural numbers, integers, etc.).
Roster Notation
Roster notation is a way to denote a set by listing its members inside curly braces.
The general form: open curly brace, list of elements separated by commas, …, end curly brace. Example: a representation of the lowercase letters in the English alphabet, typically shown as {a, b, c, d, e, f, g, …, z}.
Visual cues: open curly brace ("{") starts the set, end curly brace ("}") ends it.
Ellipsis: the "…" indicates the pattern continues without listing every element explicitly.
When to use roster notation: useful for sets with a small number of objects or where the pattern is easily enumerated.
Demonstration example from the transcript:
A simplified roster for lowercase letters shown as {a, b, c, e, f, g} with the understanding that the pattern continues up to z (the transcript shows a partial listing).
Conceptual example: the set of natural numbers is infinite and can be described in roster form only when listing a finite portion; more typically natural numbers are described with set-builder notation or interval-like descriptions.
Note on natural numbers convention:
In this class, natural numbers start at 1 (i.e., 1, 2, 3, …).
Some textbooks include 0 as a natural number; here, 0 is not included in the roster example used, but it’s acknowledged as a convention difference.
Quick recap line from the instructor: roster notation is a useful way to denote a set when you can clearly list its members or establish a simple pattern.
Set Builder Notation
Set-builder notation describes the properties that all members of the set must satisfy, rather than listing members explicitly.
General form: { x ∈ S | property }, or more compactly { x ∈ S : property } where S is a universal set or type (e.g., natural numbers).
Reads as: the set of all x in S such that [property] holds.
The transcript emphasizes two key parts:
The left side defines the domain/type of elements (e.g., x ∈ natural numbers).
The vertical bar (|) is read as “such that” and introduces the condition that must hold for x to be in the set.
Worked example from the transcript:
Problem: The set of natural numbers that are greater than 2 and at most 5.
Rostering the same set:
{3, 4, 5}
Set-builder form:
{ x ∈ \mathbb{N} \mid x > 2 \text{ and } x ≤ 5 }
The expression reads: all x in the natural numbers such that x > 2 and x ≤ 5.
Key phrases:
"x ∈ natural numbers" indicates the type of elements in the set.
The vertical bar means "such that" and introduces the conditions.
The properties are what qualify an object to be in the set; anything not satisfying them is not in the set.
Why use set-builder notation: it’s more flexible for describing potentially infinite or large sets where listing elements is impractical.
Interval Notation
Interval notation is particularly useful for describing subsets of the real numbers.
It’s a concise way to specify ranges of real numbers, often used after introducing real-number sets and their order.
The instructor notes that interval notation will be especially relevant given that the course primarily deals with real numbers.
While the transcript does not provide explicit interval examples, the emphasis is on its usefulness for real-number sets.
Unions and Intersections
Beyond single sets, we can combine sets to form new sets.
The two operations highlighted as especially useful in this course are:
Unions: combining elements that belong to at least one of several sets.
Intersections: combining elements that belong to all of several sets.
The transcript notes that there are other possible operations for forming new sets, but unions and intersections are the ones that will be most useful for this class.
Domain and Range: Two Ways to Think About It
The main topic of the section is domain and range, and two complementary perspectives are introduced: 1) Graphical analysis: determine the domain and range by looking at a graph of a relation or function.
How to tell domain from a graph: identify all x-values that appear on the graph (the x-coordinates of points on the graph); these x-values form the domain.
How to tell the range from a graph: identify all y-values that appear on the graph (the y-coordinates of points on the graph); these y-values form the range.
2) Algebraic/function-based analysis: determine domain and range by analyzing a function's formula.Example function introduced:
For domain and range, we’ll analyze the function to see what x-values are allowed and what y-values can be obtained.
The example function is used to illustrate the process of finding domain and range in a concrete way, and to connect the discussion of domain and range to a specific function.
The two approaches are complementary:
Graphical approach gives an immediate visual understanding.
Algebraic approach (via the formula) provides precise conditions (such as restrictions from denominators, square roots, etc.).
The instructor indicates this dual approach will be the focus for the course's treatment of domain and range.
Quick Rosters and Natural Numbers (Roster and Notation Review)
Roster notation recap: another quick review of how to denote a set by listing members with curly braces.
Example with letters (lowercase) of the English alphabet:
Open curly brace, list elements, end curly brace, and use ellipsis to indicate continuation: {a, b, c, e, f, g, …, z} (as shown in the transcript; pattern continues).
This demonstrates listing a subset of elements from an alphabet sequence or pattern.
Open and end curly braces are described explicitly for clarity:
Open curly brace: {
End curly brace: }
The transcript reinforces the idea that roster notation is practical for sets with a clear finite list or easily described pattern.
Natural numbers: recap of the roster idea for an infinite set, noting that natural numbers are positive integers and the set can be described by a pattern rather than a finite listing:
In this lesson: natural numbers start at one: 1, 2, 3, …
Some textbooks include 0 in the natural numbers; the instructor notes this as a convention depending on the textbook.
Overall: The roster approach can illustrate the concept of an infinite set if a pattern is recognized, but for infinite sets, set-builder notation or interval-like descriptions are typically used in practice.
Example Function and Notation Highlights
Function example used for domain/range discussion:
This form is chosen because it highlights a common domain restriction (division by zero).
When determining domain: exclude x = 5 to avoid division by zero.
When determining range (in standard real-valued function context): exclude y = 0 (the function cannot take the value 0 for any real x).
The notation used:
Natural numbers membership:
Set description with condition and membership: { x \in \mathbb{N} \,|\, x > 2 \ \text{ and }\ x \le 5 }
The vertical bar inside set-builder notation is read as "such that".
Quick reference to notation terms used in the lesson:
Membership symbol: (element of)
Such that: the vertical bar "|"
Real numbers context: often described with interval and interval notation in subsequent material
Quick References (Memory Aids)
Curly braces indicate a set; the contents can be listed (roster) or described by properties (set-builder).
Rostering is convenient for finite or pattern-easily-enumerated sets.
Set-builder notation generalizes nicely to infinite or large sets, especially when the property cleanly defines membership.
For domain and range: use graphs or formulas to determine allowable x-values (domain) and the corresponding y-values (range).
In real-number contexts, interval notation will be a key tool for expressing sets of numbers on the real line.