SetsSets

SCIE CentreSCIE Centre Additional Maths Additional Maths

© Adam Gibson

Aims:

•To understand the idea of a set

•To be able to use the appropriate mathematical symbols (such as ) to describe sets

•To be able to use Venn diagrams and make calculations

A set is any collection of distinct objects.Give me FOUR members of each of these sets:

: 3 8,C x x x

: 3 ,A x x y y

B = All natural numbers which are a multiple of 2 but not a multiple of 4

Can you tell me B A C

SETSSETS

? {6}D

SET NOTATIONSET NOTATION

Say these aloud:

A B

A B

A BA B

x A

x A

( )n A

'A

A union B

A intersection B

A is a proper subset of B

A is a subset of B

x is not a member of A

The number of elementsof A

The complement of A

The null set

The universal setx is a member of A

DESCRIBING SETSDESCRIBING SETS

: 3 8,C x x x

{3,4,5,6,7}C

These are the ELEMENTS or MEMBERS of C

Some Basic Definitions

Definition of a SetWe define a set as a collection of objects with the property that, given an arbitrary object, it is possible to tell whether or not that object belongs to the set.

Definition - Equality of SetsTwo sets A and B are said to be equal, written A = B, if they have the same elements.

Definitions – SubsetIf A and B are sets, B is said to be a subset of A if every element of B is also an element of A. That is, B ⊆ A if x ∈ B ⇒ x ∈ A

{apple, orange, banana} = {orange, banana, apple}

Historical Aside

Bertrand Russell tried to formaliseMathematics based on logic.However, he came across a problem…

Is the set of all sets which are not membersof themselves a member of itself?

“Russell’s paradox”

1+1=2

CONCEPT CHECK …CONCEPT CHECK …

•What is ?( )n A: The null set has NO elements, so the answer is zero.

•What is ?

A: The number of elements in the universal set will depend on the problem (often it will be infinite).

( )n

•True or false?

A: True. The null set is a proper subset of any other set, by definition.

C

In the box is every student in the school.

M for kids in your MATHS classS for kids in your SCIENCE class

G for kids in your GYM class

Students in Math OR Science Students in Math AND Science

Students in Math AND Science AND Gym

Students NOT in Gym or Math

SM SM

GSM ( ) 'M G

Students NOT in Gym or Math Students NOT in Gym AND Math

In other words: you start to tell a joke in math class, but the bell ringsAnd you have to finish it in gym.

Who DOESN’T get the joke?These guys heard the whole joke

Sooo everyone else

( ) 'M G ( ) 'M G

A B C

D E F

ABCDEF

Answers

( ) 'M S

'M S

'M SSM

'M S

' 'M S

IS SET THEORY USEFUL?IS SET THEORY USEFUL?

QuestionOf the 200 candidates who were interviewed for a position at a call center, 100 had a two-wheeler, 70 had a credit card and 140 had a mobile phone.

40 of them had both a two-wheeler and a credit card.30 had both a credit card and a mobile phone.60 had both a two wheeler and mobile phone. 10 had all three.

How many candidates had none of the three?

NOTEWORTHY RESULTSNOTEWORTHY RESULTS

Hence, solve the problem and draw a Venn diagram

( ) ' ' '

( ) ' ' '

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

A B A B

A B A B

A B C A B C

A B C A B C

n A B n A n B n A B

n A B C n A n B n C

n A B n A C n B C n A B C

SOLUTIONSOLUTION

all applicants

T

MC

T = two wheelersM = mobile phonesC = credit cards

( ) 200n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

n T M C n T n M n C

n T M n T C n M C n T M C

SOLUTIONSOLUTION

QuestionOf the 200 candidates who were interviewed for a position at a call center, 100 had a two-wheeler, 70 had a credit card and 140 had a mobile phone. 40 of them had both a two-wheeler and a credit card.30 had both a credit card and a mobile phone.60 had both a two wheeler and mobile phone. 10 had all three.

( ) 100, ( ) 140, ( ) 70n T n M n C

( ) 40, ( ) 30, ( ) 60n T C n M C n T M

( ) 10n T M C

(( ) ') 200 100 140 70 (40 30 60) 10n T M C

So there are 10 job applicants with none of the three.

PRACTICE TASKSPRACTICE TASKS

1 Convert these English statements to set notationa) The number of elements in C AND D is 24b) The number of elements in either A or B is 5.c) Set X is the intersection of sets Y and Zd) Sets A and B have no common elementse) The number of elements in neither A nor B is 1f) G is not a proper subset of H

2) a) List all subsets of A = {5,6,9} b) How many subsets does B have, B = {x:x<20, x is prime}

3) Let k, x and y all be natural numbers. We define S(k) as the set of number pairs (x,y) as follows:

( ) {( , ) : }S k x y x y k Plot a graph of n(S(k)) against k. Can you find the equationfor n(S(k)) as a function of k?

4) Cantor says that the number of elements in the set{1,2,3,4…} is the same as the number of elements in the set {2,4,6,8…}. Is he right?