1 2 Sets and Functions

The symbol means ‘is an element of’.

Introduction to Set TheoryIn Mathematics, the word set refers to a group of numbers or other types of elements. Sets are written as follows:

Examples

{ 1, 2, 3, 4,

5, 6 }{ -0.7, -0.2, 0.1

}{ red, green,

blue }

4 { 1, 2, 3, 4, 5 }

7 { 1, 2, 3 }

{ 6, 7, 8 } ⊆ { 6, 7, 8, 9 }

N

W

Z

{ 1, 2, 3, 4, 5,

... }{ 0, 1, 2, 3, 4, 5,

... }{ ... -3, -2, -1, 0, 1, 2,

3, ... }

The Basic Number Sets

QRational numbers Includes all integers, plus any number which can be written as a fraction.

R √7 π Includes all rational numbers, plus irrational numbers such as or .

Real numbers

Whole numbers

Integers

Natural numbers

N

WZ

Q

R

√82

57

Set Theory and Venn DiagramsVenn Diagrams are illustrations which use overlapping circles to display logical connections between sets.

The ‘input’ is also known as the domain of the function, with the ‘output’referred to as the range.

Functions and Mappings

f (x)domain range

Important

Each number in the domain has one output number in the range.

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 inthe formula.

INPUT OUTPUT

A•1•2•3

B•3•7•9

Domain Range

This is known as an arrow diagram.

Functions and Arrow Diagrams

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.

A typical graph of a function f shows the points (a, f(a)) for all values x = a in the domain of f.

Function Graphs

range(‘out’ numbers)

domain(‘in’ numbers)

f(a)

a

(a, f(a))

y = f(x)

How can I decide from a graph whether it is a function?

The Vertical Line Test

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