ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Coordinate Line, Sets, and Intervals (Curious Concepts Precalculus 1.5)

Updated on August 28, 2013

Self-explain Learning Technique

Self-explain Learning Technique help people to better comprehend and remember any thing they want to learn. It my sound strange, but it is many times more effective then re-reading. To use this technque, give short answers to the prompted questions. The more privious knowlege that you incorperate into your answer, the more effective this learning technique is. On average it improves learning by over 2.5 standard deviations; the difference between a C student and an A student.

How could a solid understanding of math's basic structure help you?

The Coordinate Line has an Origin at the point of 0 (kinda coincidental huh). All real numbers can be plotted on it. From this Origin at 0: Natural Numbers ascend to the right, negative Integers descend to the left, Rational and Irrational numbers are plotted at their true-or-approximate values.

Coordinate Line

There is an inherent order to the coordinate line that can be expressed by relationships. We can say that a is greater then b by writing a>b if a-b>0; that is a minus b is greater then 0. We can say that a is less then b by writing a<b if a-b<0. You should be, or soon get, very familiar with this notation; but I still use the alligator always eating the bigger number mnemonic when things get complicated. The greater then sign can be modified to create ≥, meaning equal to or greater then. The less then sign can be modified to create ≤, meaning equal to or less then.

the aligator eats the bigger number, greater then and less then mnemonic.
the aligator eats the bigger number, greater then and less then mnemonic.

Self-explain Learning Technique

If you had a pocket full of change; you could concider each denomination an set, and the year the coin was made an element. How could you apply sets and elements to more then chump change?

Sets

A collection of objects (numbers) is called a Set, and each object (number)is call an Element. The element symbol, ε, is used to express this relationship. a ε B is read as a is an element of B.

What would you expect the meaning of the element symbol with a line through it would mean? It means "not an element", as in meaning c is not an element of D.

Sets can be made by listing elements inside of braces.

Example: K={2,3,4,5,6}

We can also use K in a set-builder notation. correspondingly

Example: K={x│x is an integer and 1<x<7}

This is read: K is the set of all x; such thatx is an integer and 1 is less thanx, and x is less then 7.

Unions are a collection of all elements in more then 1 set, and the relationship is expressed by the symbol. If G and R are Sets, then there Union includes all elements that are in G, and R, or both.

A Union of pionts that appear in both Sets, G and R is writen as: G U R

Intersections are were two or more sets over lap, and are expressed with the symbol ∩. If L and M are Sets, then there intersection includes only elements that are found in both L and M.

If you want express the objects that only appear in both sets L and M , you could write: LM


Unions include all the objects in both sets. An object only needs to fall into one of the sets being combined to be included. Intersections are were the sets overlap, an object needs to be part of both sets.

Self-explain Learning Technique

How would you compare rocks and hard places to intervals?

Intervals

Intervals are sets of real numbers and are very common in calculus. Intervals correspond to line segments.

The Open Interval between two points, points a and b,are shown as open circles at their points on a graph. a < b, is denoted with parentheses (a,b), and consists of all possible points between a and b, but not a and b.

The Closed Interval includes the end points a and b , a ≤ b, along with all the points between them; a ≤ b. Closed intervals are denoted with brackets: [a,b]. The points a and b are shown as filled circles at their points on a graph.

Using set-builder notation, we write:

(a,b) = {x│a<x<b} [a,b] = {x│a≤x≤b}

They are shown as on a line graph:


(a,b)

<---------------------o================o------------------------->


[a,b]

<--------------------------●==================●---------------->


The Combined Interval uses both, parentheses and brackets, to show that one end point is included but the other is not.

If only one end point exists, the infinity symbols, ∞ or -∞, can be used to show that a set continues to infinitely in one direction.


working

This website uses cookies

As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://corp.maven.io/privacy-policy

Show Details
Necessary
HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
LoginThis is necessary to sign in to the HubPages Service.
Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
AkismetThis is used to detect comment spam. (Privacy Policy)
HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
Features
Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
MavenThis supports the Maven widget and search functionality. (Privacy Policy)
Marketing
Google AdSenseThis is an ad network. (Privacy Policy)
Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
Index ExchangeThis is an ad network. (Privacy Policy)
SovrnThis is an ad network. (Privacy Policy)
Facebook AdsThis is an ad network. (Privacy Policy)
Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
AppNexusThis is an ad network. (Privacy Policy)
OpenxThis is an ad network. (Privacy Policy)
Rubicon ProjectThis is an ad network. (Privacy Policy)
TripleLiftThis is an ad network. (Privacy Policy)
Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
Statistics
Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)
ClickscoThis is a data management platform studying reader behavior (Privacy Policy)