Learning Objectives
Introduction
(pi) belong to. Some of them belong to
more than one set. I think you are ready to go forward. Let's
make you a numeric set whiz kid (or adult).
Tutorial
I feel sorry for 0, it does not belong to either group. It is
neither a positive or a negative number.
When graphing a point on the number line, you simply color in a point that corresponds to that number on the number line as illustrated below. That is how you graph a solution on the number line.
This is how you would graph it if your solution was the number 2:
Sets and Elements
Those objects are generally called elements of the set.
The symbol 
means 'is an element of.'
So, it stands to reason that 
represents 'is not an element of.'
Subset
B, when every element of A is contained in B.
(It does not necessarily mean that every element of B is also contained
in A)
Ways to Notate Sets
{January, June, July}
Set builder notation describes the members of the set without listing
them. It is also written between two set brackets.
For example,
{x | x is a month that begins with J}
When writing it in set builder notation you always do the following: start off with a left set bracket, then you put x followed by a vertical bar which is interpreted as 'such that'. Then you write out the description of the elements of the set. Finish it with a right set bracket.
So the above illustration would be read: "x, such that, x is a month that begins with J."
It is important to know set builder notation, especially in mathematics, because it allows you to group together large number of elements that belong to a certain category. The above set has only 3 elements, so it would not be difficult to write it in roster form as shown above. However, if your set has hundreds or thousands of elements, it would be hard to list them out, but easy to refer to them using set builder notation. For example, {x| x is a college student in Texas}.
Before we move on to the math aspect of sets, there is one more term
we need to make sure you have a handle on.
Empty Set
It is symbolized by { } OR 
.
Be careful. It is real tempting to use them together, but {}
IS NOT a way to indicate empty set.
Let's move on to some special sets that pertain specifically to math.
Note that the three dots shown in the sets below are called ellipsis.
It indicates that the elements in the set would continue in the same
pattern. - In other words, the list would keep going and going in that
direction using the pattern illustrated.
N = {1, 2, 3, 4, 5, ...}
{0, 1, 2, 3, 4, 5, ...}
Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4,
5, ...}
The natural numbers and the whole numbers are both subsets of integers.
Q = {
|
a and b are integers and 
}
Be very careful. Remember that a whole number can be written as one integer over another integer. The integer in the denominator is 1 in that case. For example, 5 can be written as 5/1.
The natural numbers, whole numbers, and integers are all subsets
of rational numbers.
I = {x | x is a real number that
is not rational}
One big example of irrational numbers is roots of numbers that are
not perfect roots - for example 
or 
.
17 is not a perfect square - the answer is a non-terminating,
non-repeating decimal, which CANNOT be written as one integer over another.
Similarly, 5 is not a perfect cube. It's answer is also a non-terminating,
non-repeating decimal.
Another famous irrational number is
(pi). Even though it is more commonly known as 3.14, that is
a rounded value for pi. Actually it is 3.1415927... It would keep
going and going and going without any real repetition or pattern. In other
words, it would be a non terminating, non repeating decimal, which again,
can not be written as a rational number, 1 integer over another integer.
R = {x | x corresponds to point
on the number line}
Example
1: List the elements of the set {x | x is a whole number less than 11}
Putting these two ideas together we get:
{0, 1, 2, 3, ..., 9, 10}
Example
2: List the elements of the set {x | x is a natural number between 5 and 11}.
Let's see what we get when we put those ideas together:
{6, 7, 8, 9, 10}
You would not have an ellipsis after the 10 because this set would stop
at the number 10.
Example
3: List the elements of the set {x | x is a natural number greater than 100}.
Let's see if you figured this one out ok:
{101, 102, 103, 104, ...}
Did you remember to include the ellipsis to show that the set would
continue on and on in the same pattern?
Example
4: Graph the set on a number line.
{-2, -.5, 0, 1/5, 3}
In this problem, we have a -.5, which is between -1 and 0. Since it is halfway between these two numbers, I would place the dot halfway between.
We also have the fraction 1/5, which is between 0 and 1 and since it is closer to 0 than 1, I would place it accordingly on the graph.
The other numbers are integers that are already marked clearly on the graph.
Let's see what we get when we graph all of these real numbers:
Example
5: List the elements of the following sets that
are also elements of the given set
{-4, 0, 2.5, , 
, 
,
11/2, 7}
Natural numbers, whole numbers, integers, rational numbers, irrational
numbers, and real numbers.
{, 7}.
Note that simplifies
to be 5, which is a natural number.
{0, , 7}.
{-4, 0,, 7}.
{-4, 0, 2.5, , 11/2,
7}.
{, }.
These two numbers CANNOT be written as one integer over another.
They are non-repeating, non-terminating decimals.
{-4, 0, 2.5, , ,,
11/2, 7}.
Example
6: Place a or to make the statement true.
0 ? {x | x is
a whole number}
0
{x | x is a whole
number}.
Example
7: Place a or to make the statement true.
-2 ? {2, 4, 6, .... }
-2 {2,
4, 6, .... }.
Example
8: Place a or to make the statement true.
½ ? {x | x is an irrational number}
½ {x | x is an irrational
number}.
Example
9: Place a or to make the statement true.
?
{x | x is a rational
number}
simplifies
to be 2, which can be written as 2/1, we can write the following true statement
{x | x is a rational
number}.
In fact, there are no elements in N that
are in I.
Well, let me tell you why!
The absolute value of x, notated |x|, measures the DISTANCE that x is away from the origin (0) on the real number line.
Aha! Distance is always going to be positive (unless it is 0)
whether the number you are taking the absolute value of is positive or
negative.
The following are illustrations of what absolute value means using the numbers 3 and -3:
Example
13: Find the absolute value.
|-7|
I came up with 7, how about you?
Example
14: Find the absolute value. |7|I came up with 7, how about you?
Example
15: Find the absolute value.
- |-2|First of all, if we just concentrate on |-2|, we would get 2.
Second, note that there is a negative on the OUTSIDE of the absolute value. That means we are going to take the opposite of what we get for the absolute value.
Putting that together we get -2 for our answer.
Note that the absolute value part of the problem was still positive.
We just had a negative on the outside of it that made the final answer
negative.
Opposites
The opposite of x is the number -x.
Keep in mind that the opposite of 0 is 0.
The following is an illustration of opposites using the numbers 3
and -3:
Double Negative Property
When you see a negative sign in front of an expression, you can think
of it as taking the opposite of it. For example, if you had -(-2),
you can think of it as the opposite of -2. Since a number can only
have one of two signs, either a '+' or a '-', then the opposite of a negative
would have to be positive. So, -(-2) = 2.
Example
16: Write the opposite of 1.5.
Example
17: Write the opposite of -3.
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1c: List the elements of each set.
1b. {x | x is a whole number between 100 and 105}
(answer/discussion
to 1b)
,
}
Need Extra Help on these Topics?
http://www.purplemath.com/modules/numtypes.htm
This website goes over the different types of numbers: natural, whole,
integers, rational, irrational, and real.
http://www.mathleague.com/help/integers/integers.htm#positiveandnegativeintegers
This webpage will help you with positive and negative integers.
http://www.mathleague.com/help/integers/integers.htm#thenumberline
This webpage will help you with the number line.
http://www.mathleague.com/help/integers/integers.htm#comparingintegers
This webpage will help you with comparing two integers.
http://www.purplemath.com/modules/absolute.htm
This webpage will help you with absolute value.
http://www.mathleague.com/help/integers/integers.htm#absolutevalueofaninteger
This webpage will help you with absolute value of an integer.
http://www.math.utah.edu/online/1010/line/
This webpage will help you with ordering of integers and absolute value.
http://www.eduplace.com/math/mathsteps/7/b/
This webpage helps you with absolute value.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on June 10, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.