1. Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004

2. Topic 1 – Number and Algebra
1.1 – The Number Sets

3. Section A – Some Set Language
A set is a collection of numbers or objects.
- If A = {1, 2, 3, 4, 5} then A is a set that contains those numbers.
An element is a member of a set.
- 1,2,3,4 and 5 are all elements of A.
-  means ‘is an element of’ hence 4  A.
-  means ‘is not an element of’ hence 7  A.
-  means ‘the empty set’ or a set that contains no elements.

4. Subsets If P and Q are sets then: P  Q means ‘P is a subset of Q’.
Therefore every element in P is also an element in Q.
For Example:
{1, 2, 3}  {1, 2, 3, 4, 5} or
{a, c, e}  {a, b, c, d, e}

5. Union and Intersection
P  Q is the union of sets P and Q meaning all elements which are in P or Q.
P ∩ Q is the intersection of P and Q meaning all elements that are in both P and Q.
A = {2, 3, 4, 5} and B = {2, 4, 6}
A  B =
A ∩ B =

6. M = {2, 3, 5, 7, 8, 9} and N = {3, 4, 6, 9, 10}
Is:
M  N ?
{9, 6, 3}  N?
True or False?
4  M
6  M
List:
M ∩ N
M  N

7. Section B – Number Sets Reals R Q Rationals Integers Natural Z
Irrationals
(fractions; decimals that repeat or terminate)
(no fractions; decimals that don’t repeat or terminate)
Integers
(…, -2, -1, 0, 1, 2, …) Z
Natural
(0, 1, 2, …) N

8. Section B – Number Sets
N = {0, 1, 2, 3, 4, …} is the set of all natural numbers.
Z = {0, + 1, + 2, + 3, …} is the set of all integers.
Z+ = {1, 2, 3, 4, …} is the set of all positive numbers.
Z- = {-1, -2, -3, -4, …} is the set of all negative numbers.
Q = { p / q where p and q are integers and q ≠ 0} is the set of all rational numbers.
R = {real numbers} is the set of all real numbers. All numbers that can be placed on a number line.

9. - Show that 0.45 and 0.88888… are rational -

10. Topic 1.1 Summary
Sets      N Z Q R

11. Topic 1.1 Summary Sets  - is an element of  - is not an element of
 - is a subset of
 - union (everything)
- intersection (only what they share)
N – natural numbers
Z - integers
Q – rational numbers
R – real numbers