How To Graph Set Builder Notation

Learning Outcomes

• Write sets using set up-builder, inequality, and interval notation
• Depict sets on the real number line using gear up builder, interval, and inequality notation

There are several ways to define sets of numbers or mathematical objects. The reason we are introducing this here is considering we often need to define the sets of numbers that brand up the inputs and outputs of a role.

How we write sets that make up the domain and range of functions frequently depends on how the relation or role are divers or presented to united states of america.  For instance, we tin can use lists to describe the domain of functions that are given as sets of ordered pairs. If we are given an equation or graph, nosotros might use inequalities or intervals to describe domain and range.

In this department, we will introduce the standard notation used to define sets, and give you a chance to practice writing sets in three means, inequality annotation, gear up-builder notation, and interval notation.

Consider the set [latex]\left\{x|10\le x<xxx\right\}[/latex], which describes the beliefs of [latex]10[/latex] in set-architect notation. The braces [latex]\{\}[/latex] are read every bit "the set of," and the vertical bar [latex]|[/latex] is read as "such that," so we would read [latex]\left\{x|ten\le 10<30\right\}[/latex] as "the gear up of x-values such that x is less than or equal to [latex]x[/latex], and [latex]10[/latex] is less than thirty."

The table beneath compares inequality note, set-builder notation, and interval notation.

	Inequality Annotation	Set-builder Notation	Interval Notation
	[latex]5<h\le10[/latex]	[latex]\{h | v < h \le x\}[/latex]	[latex](5,x][/latex]
	[latex]5\le h<ten[/latex]	[latex]\{h | five \le h < 10\}[/latex]	[latex][v,ten)[/latex]
	[latex]5<h<10[/latex]	[latex]\{h | five < h < ten\}[/latex]	[latex](5,ten)[/latex]
	[latex]h<10[/latex]	[latex]\{h | h < 10\}[/latex]	[latex](-\infty,ten)[/latex]
	[latex]h>10[/latex]	[latex]\{h | h > 10\}[/latex]	[latex](10,\infty)[/latex]
	All real numbers	[latex]\mathbf{R}[/latex]	[latex](−\infty,\infty)[/latex]

To combine ii intervals using inequality note or prepare-builder notation, nosotros use the give-and-take "or." Every bit nosotros saw in earlier examples, nosotros utilize the matrimony symbol, [latex]\cup [/latex], to combine two unconnected intervals. For example, the union of the sets [latex]\left\{2,3,5\correct\}[/latex] and [latex]\left\{4,6\right\}[/latex] is the prepare [latex]\left\{2,3,4,5,6\right\}[/latex]. It is the set of all elements that belong to one or the other (or both) of the original ii sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending club of numerical value. If the original two sets have some elements in common, those elements should be listed but once in the union set. For sets of real numbers on intervals, some other instance of a union is

[latex]\left\{x|\text{ }|x|\ge 3\correct\}=\left(-\infty ,-3\correct]\cup \left[3,\infty \correct)[/latex]

This video describes how to apply interval notation to describe a set.

This video describes how to apply Set-Builder notation to describe a set.

A General Note: Ready-Builder Annotation and Interval Notation

Set-builder note is a method of specifying a set of elements that satisfy a sure condition. It takes the form [latex]\left\{10|\text{statement about }x\correct\}[/latex] which is read as, "the set of all [latex]10[/latex] such that the statement most [latex]x[/latex] is true." For case,

[latex]\left\{x|4<10\le 12\correct\}[/latex]

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may non exist included and an upper limit that may or may not exist included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the ready. For case,

[latex]\left(four,12\correct][/latex]

How To: Given a line graph, describe the set of values using interval annotation.

1. Identify the intervals to exist included in the gear up by determining where the heavy line overlays the existent line.
2. At the left end of each interval, apply [ with each end value to be included in the fix (solid dot) or ( for each excluded end value (open dot).
3. At the right end of each interval, use ] with each end value to be included in the fix (filled dot) or ) for each excluded end value (open dot).
4. Utilise the wedlock symbol [latex]\cup [/latex] to combine all intervals into one set.

Example: Describing Sets on the Real-Number Line

Describe the intervals of values shown beneath using inequality note, gear up-builder notation, and interval notation.

Endeavor Information technology

Given the graph beneath, specify the graphed set in

1. words
2. set-architect notation
3. interval annotation

The table below gives a summary of interval notation.

Contribute!

Did you lot accept an thought for improving this content? We'd love your input.

Meliorate this pageLearn More

Source: https://courses.lumenlearning.com/waymakercollegealgebra/chapter/standard-notation-for-domain-and-range/

Share this post