Category Archives: Resources

Getting Started, How-Tos, and so on

Homework Hints: Intervals of real numbers – unions and intersections

Hi everyone,

This is a followup to Thursday’s lecture, and should provide a little help with some of the homework problems (I’m looking at you, Problem 7).

Example.  Consider the intervals of real number A = [2, 5) and B = (4,\infty).  Find their intersection A\cap B and their union A\cup B.

One key idea is that these are intervals of the real numbers, so they include not just the whole numbers but all numbers between the endpoints.  The set A includes all numbers that are great than or equal to 2 and less than 5.  This means that A includes 2, 3 and 4, but also decimals such as 3.5 or 4.9998.  The set B includes all numbers greater than 4, such as 4.1 or six billion.

The intersection will be the places where these two overlap Рit will include numbers greater than 4 but less than 5 (NOTE: it does not include the numbers 4 and 5 themselves, but it does include, for example, 4.3).  In interval notation, we write:

  • A\cap B = (4,5)

The union will include all numbers greater than or equal to 2, written:

  • A\cup B = [2,\infty)

WeBWorK Tip: ¬†To enter the infinity symbol, just use the word “infinity” like this:
[2, infinity)

WeBWorK Tip: Sometimes in WeBWorK, your answer will consist of two different intervals Рyou want to include them both in the answer.  To do this, connect them with a union symbol (just use the capital U on your keyboard).  Here is a (made up) example:
[1,7] \cup (15,17]

Not sure if these will help, but they may give you a little more to go on – feel free to leave a comment here or send me an email if you have questions.

Best of luck!

Prof. Reitz

Getting Started with WeBWorK

WeBWorK is accessible from on and off campus (anywhere you have access to the internet).  Your first two WeBWorK assignments are due on Tuesday, September 9th, at midnight, and will cover the material from the first two weeks of class (primarily from the second week).  Here’s what you have to do:

Assignment.  To get started , you must complete the following three steps.

Step 1.  Log in to WeBWorK here: http://mathww.citytech.cuny.edu/webwork2/MAT2071-F15-Reitz/.  I have created Usernames and Passwords for each student registered for my class.

Username. ¬†Your username for WeBWorK consists of your first initial plus your last name, all lowercase (for example, John Smith would have username ‚Äėjsmith‚Äô).

Password. ¬†Your temporary password is the same as your username (if your username is ‘jsmith’, your password is currently ‘jsmith’).

Step 2. ¬†Change your password and update your email address. ¬†To do this, select ‚ÄúPassword/Email‚ÄĚ from the main menu on the left. ¬†Use whatever email address you like (I suggest using one that you check often).

Step 3.  Complete the first two assignments, titled Assignment1-Sec1.2-1.3 and Assignment2-Sec1.4-1.7. Click on an assignment on the main screen to get started.

If you have¬†any trouble¬†‚Äď either with logging in, or with completing the assignment, post a comment here or send me an email and I will get back to you.

WeBWorK Tips:

  1. Click on a problem to see the details (the list of problems appears in the menu on the left). ¬†Enter an answer and hit ‚ÄúSubmit Answers‚ÄĚ. ¬†Don‚Äôt worry, if you get it wrong you can try it again.
  2. You can work on the problems in any order you wish.  You can do some problems now, and come back and do the rest another day (your work will be saved, as long as you submit your answers).
  3. If you want to print out a copy of the assignment, click on the assignment name in the main menu on the left, and then click the link in the main screen area that reads ‚ÄúDownload a hardcopy of this homework set.‚ÄĚ

Homework Hints: Set-theoretic notation

Hi everyone,

I wanted to give an example of writing a given set in set-theoretic notation Рthis should help out with some of the problems on your first homework assignment (Section 1.1).

Example: Write in set-theoretic notation:  \{5,10,15,20,\ldots \}

In this case, you can see that the given set consists of all multiples of 5.  A good way to approach problems like this is to start with one of the basic sets, for example

  • ¬†the natural numbers \mathbb{N} = \{1,2,3,4, \ldots \}
  • the integers \mathbb{Z} = \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}

In this example, I can see that multiplying every natural number by 5 should give me the set that I want.  Therefore, I will use the formula 5n , and the condition n\in \mathbb{N}.  Combining these in set-theoretic notation gives the solution:

Solution \{ 5n : n\in \mathbb{N} \}

Here are two ways to read this solution aloud:

  • “the set of all 5n such that n is a natural number”, or
  • “for each n in the natural numbers, multiply n by 5 and include the result in the set”

I hope this helps Рfeel free to respond here if you any questions.

Best,
Prof. Reitz