SET OPERATIONS AND TYPES OF
NUMBERS
Objectives:
1. Student should be able to discuss, describe and use the concepts in this worksheet.
2. Students should have a sound understanding and ability to use roster, set builder and interval
notation and describe their differences.
Target:
MAT0511 access mathematics students; any student needing to work on set operations and types of
numbers.
Facilitation:
Quiz Workshop. Students sat in a circle passing around a bucket containing the list of words/concepts
below. These concepts/words were discussed either graphically, mathematical notation and/or
described in words.
Define The Following:
Set.............................................................................................................................................
Inequalities...............................................................................................................................
Union........................................................................................................................................
Number Line.............................................................................................................................
Proper Subset...........................................................................................................................
Algorithm.................................................................................................................................
Interval.....................................................................................................................................
Recurring Decimal.....................................................................................................................
Positive Number.......................................................................................................................
Set Builder Notation.................................................................................................................
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Half Closed Intervals.................................................................................................................
Half Open Intervals...................................................................................................................
Open Interval.........................................................................................................................
Closed Intervals......................................................................................................................
Irrational Numbers..................................................................................................................
How would an empty set be represented?...............................................................................
Integers...................................................................................................................................
Non–positive Number.............................................................................................................
Disjoint Set..............................................................................................................................
Roster Notation.......................................................................................................................
Decimal....................................................................................................................................
Magnitude................................................................................................................................
Terminating Decimal.................................................................................................................
What should be avoided when using INEQUALITIES? ................................................................
.................................................................................................................................................
Finite.........................................................................................................................................
At least.....................................................................................................................................
Infinite......................................................................................................................................
Intersection...............................................................................................................................
Variable.....................................................................................................................................
Digit..........................................................................................................................................
Venn diagram...........................................................................................................................
Negative Number.....................................................................................................................
How do you read an Inequality? (Hint: Direction)...................................................................
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At Most.....................................................................................................................................
Not More than..........................................................................................................................
Natural Numbers......................................................................................................................
Real Numbers...........................................................................................................................
Ordinal.....................................................................................................................................
Rational Numbers....................................................................................................................
Whole Numbers........................................................................................................................
Non-Negative Number.............................................................................................................
What are the differences between roster, interval and set builder notation..........................
..................................................................................................................................................
Draw A
3;2 and B
0;5 on separate number lines, but underneath the main number line,
so that if there is a 2 in set A and a 2 in set B they are beneath each other. Draw A B and
A B on separate number lines too. Now use the graphs to write A B and A B in set
builder notation beginning with interval notation
A
3;2
0
B
0;5
0
A
B
0
A
B
0
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..........................................................................................................................................................
..........................................................................................................................................................
..........................................................................................................................................................
..........................................................................................................................................................
..........................................................................................................................................................
..........................................................................................................................................................
..........................................................................................................................................................
..........................................................................................................................................................
What is the difference between roster, interval and set builder notation?.................................
......................................................................................................................................................
..........................................................................................................................................................
What is the difference between using  and Z ?.........................................................................
......................................................................................................................................................
From the workshop (March 2012) these are some of the learning outcomes:



'

0



(  ' reads Q-prime. It means not rational, in other words, irrational)
Using subsets students are able to justify the bottom-up approach of the following algorithm
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 - Real Numbers
 - Rational Numbers
 ' - Irrational Numbers
 - lntegers

0
Fractions
- Whole Numbers
 - Natural Numbers
Another way to look at the above algorithm is to see the types of numbers as subsets.
 - Real Numbers (numbers found on a number line)
Non-real
Numbers
 - Rational Numbers
 ' - Irrational Numbers
p
where q
q
Decimal numbers
undefined
NON- recurring
This forms
part of
complex
numbers
 - Integers
0;
; 3; 2; 1
 0 - Whole Numbers
0
 - Natural Numbers
1;2;3;
NON -repeating
NON –terminating
i.e.
x is
3.14
e 2.718
prime number
Decimal numbers
1
0.333333 0.3 (recurring)
3
0.123123123 0. 123 (repeating)
1
2
0.25 (terminating)
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
 0 , that is, the set of natural numbers is a subset of the set of whole numbers. In fact, the
set of natural numbers is a proper subset of the set of whole numbers.
The above algorithm is an inside – out approach.
For example,
3 

7 
The notation
0 
0
3

9
0


 but 3  '

 but 7 


0
 but 0 
 but 3

9

0

- means is an element of the set;
- means is not an element of the set.
The Number line:
The number line consists of all  - real numbers.
 - real numbers form a number line.
The number line is made up of three types of numbers.
Negative Numbers
0
Positive numbers
Look what the word not does in an inequality statement:

x 3

x is not less than or equal to 3
x is not more than 3 
x 3

x is less than or equal to 3
x is more than 3
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If x 0 then x is positive (see the number line above)
If x 0 then x is non positive, that is, it is not positive (it works on the same principle as the not
in the inequality above)
Similarly,
If x 0 then x is negative (see the number line above)
If x 0 then x is non negative, that is, it is not negative
Reading an inequality from LEFT to RIGHT:
x 2
reads
x is greater than 2
2 x
reads
2 is less than x
Both mean the same thing – always read the inequality from left to right. Think of the
inequality sign as the mouth of a fish. A fish always likes to go for the biggest amount.
Remember to always have your inequality signs in the same direction.
2 x 3
reads
-2 is less than or equal to x and x is less than 3
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Graphical
representation

Graphical
representation
Interval
notation
Set builder
notation
Name
Meaning
or
or
Closed bracket
Includes the value
or
or >
Open bracket
Excludes the value
and
Intersection
Common to both;
overlapping
Set builder
notation
Name
Meaning
or
Union
Set A or set B or both
Disjoint sets
intersection is empty
Interval
notation
REFERENCES
Singleton, J. and Bohlman, C. (2009) Mathematics Access Module. Pretoria: University of South Africa.
KATE STRYDOM
March 2012
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