Transcript 1. Sets and Functions

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Sets and Functions
Introduction to Set Theory
In Mathematics, the word set refers to a group of numbers or other
types of elements. Sets are written as follows:
{ 1, 2, 3, 4, 5, 6 }
The symbol
{ -0.7, -0.2, 0.1 }
means ‘is an element of’.
Examples
4
{ 1, 2, 3, 4, 5 }
7
{ 1, 2, 3 }
{ 6, 7, 8 } ⊆ { 6, 7, 8, 9 }
{ red, green, blue }
The Basic Number Sets
N
Natural numbers
{ 1, 2, 3, 4, 5, ... }
W
Whole numbers
{ 0, 1, 2, 3, 4, 5, ... }
Z
Integers
{ ... -3, -2, -1, 0, 1, 2, 3, ... }
Q
Rational numbers
Includes all integers, plus any number
which can be written as a fraction.
R
Real numbers
Includes all rational numbers, plus
irrational numbers such as √ 7 or π .
Set Theory and Venn Diagrams
Venn Diagrams are illustrations which use overlapping circles
to display logical connections between sets.
R
5
7
Z
N
W
Q
√ 82
Functions and Mappings
What is a function?
A function, f, consists of :
1. a formula, f(x), which tells you what to do with a given value of x.
2. a domain which describes the values of x you are allowed to use in
the formula.
Important
Each number in the domain has
one output number in the range.
INPUT
domain
f (x)
OUTPUT
range
The ‘input’ is also known as the domain of the function, with the ‘output’
referred to as the range.
Functions and Arrow Diagrams
A
•1
•2
•3
B
•3
•7
•9
Domain
Range
This is known as an
arrow diagram.
Diagram 1 is a function
as each element in set A
is mapped to one and
only one element in set
B.
A
•1
•2
•3
B
•3
•7
•9
This is not a function as 2 is
mapped to 7 and 9.
Also 3 does not appear to
be mapped to anything.
Function Graphs
A typical graph of a function f shows the points (a, f(a)) for all
values x = a in the domain of f.
y = f(x)
range
(‘out’ numbers)
f(a)
(a, f(a))
a
domain
(‘in’ numbers)
The Vertical Line Test
How can I decide from a graph whether it is a function?
The vertical line test is a way to determine whether or not we have
a function.
If a vertical line intersects the graph in more than one place, then it
is NOT a function.
The test is simply a restatement of the definition of a function which
states that every x value must have a unique y value.
HHM
Exercise 2B
Page 24
Q’s 1 and 8
Success Criteria :use set notation
Illustrate vertical line test