Download Chapter 1 Number Sets and Properties

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
Topic 1 – Number and Algebra
1.1 – The Number Sets
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.
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}
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}
AB=
A∩B=
and
B = {2, 4, 6}
M = {2, 3, 5, 7, 8, 9} and N = {3, 4, 6, 9, 10}
• True or False?
I. 4  M
II. 6  M
•
List:
I. M ∩ N
II. M  N
•
Is:
I. M  N ?
II. {9, 6, 3}  N?
Section B – Number Sets
Reals
Rationals
R
Q
(fractions; decimals that repeat or terminate)
Integers
Z
(…, -2, -1, 0, 1, 2, …)
Irrationals
(no fractions;
decimals that
don’t repeat or
terminate)
 , 2, etc.
Natural N
(0, 1, 2, …)
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.
- Show that 0.45 and 0.88888… are rational -
Topic 1.1 Summary
Sets





N
Z
Q
R
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