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### Positive Numbers Less Than -3

```
Date: 01/30/98 at 14:27:27
From: Mrs. Heather
Subject: Algebra 1

Problem: Write using roster notation and set-builder notation.

1) the set C of positive multiples of 3 less than -3

How can a positive number be less than 3?  That is what's really
stumping me! Also, "negative integers greater than -3" only include
-2 and -1, right?

```

```
Date: 02/01/98 at 22:42:36
From: Doctor Jaffee
Subject: Re: Algebra 1

Greetings Mrs. Heather,

There's a good reason that you can't find any positive numbers less
than -3. There aren't any! Consequently, there are no positive
multiples of 3 less than -3.

Now, roster notation is a system in which one lists all the elements
of a set in brackets. If there are no elements in the set, then just
make 2 brackets with nothing in between them like so: {  }. (This is
called the "empty set".)

In set builder notation we want to establish some kind of rule or
equation that generates all the numbers of the set. So, if we use the
variable x to represent an integer, all the elements of the set are
numbers in the form 3 times x where 3x is less than -3, but 3x is
greater than 0 and 3x is an integer.  We would write
{3x:0<3x<-3, x an integer}. Of course, the inequality 0 < 3x < -3 has
no solution, so this notation would generate the empty set.

You were also right when you stated that -2 and -1 are the only
negative integers greater than -3.

Thanks for writing and I hope this has helped.

-Doctor Jaffee,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Sets

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