Set TheoremSet Theorem

By Derek Mok, Alex Yau, By Derek Mok, Alex Yau, Henry Tsang and Tommy LeeHenry Tsang and Tommy Lee

Definition and Info (History)Definition and Info (History)Any collection of distinct things considered Any collection of distinct things considered

as a wholeas a whole Invented at the end of the 19Invented at the end of the 19thth centaury centaurySet theory can be viewed as the Set theory can be viewed as the

foundation upon which nearly all of foundation upon which nearly all of mathematics can be built and the source mathematics can be built and the source from which nearly all mathematics can be from which nearly all mathematics can be derived derived

Element of SetsElement of SetsObject of a set = “elements” or “members”Object of a set = “elements” or “members”The element of a set can be anything:The element of a set can be anything:

NumbersNumbersPeoplePeopleLettersLettersAlphabetAlphabetOther setsOther sets

Describing SetsDescribing SetsSets can be defined in several waysSets can be defined in several ways

Defined using wordsdefined by explicitly listing its elements

between bracesWhen two description define the same set, When two description define the same set,

such as A is identical to C, we can write such as A is identical to C, we can write A=C to express this equalityA=C to express this equality

Ellipses (…) indicates that the list Ellipses (…) indicates that the list continues in an obvious waycontinues in an obvious way

Set membershipSet membership

When something IS an element of a particular When something IS an element of a particular set, then this is symbolized byset, then this is symbolized by

When something IS NOT an element of a When something IS NOT an element of a particular set, then this is symbolized byparticular set, then this is symbolized by

Example:Example: and since 285 = 17² − 4; but and since 285 = 17² − 4; but and and

Cardinality of a SetCardinality of a SetEach of the sets given above has a Each of the sets given above has a

definite number of members.definite number of members.However a set can also have However a set can also have zerozero

members, called an members, called an empty set,empty set, represented by represented by øøExample: The set Example: The set AA of all three-sided square of all three-sided square

has zero members, therefore has zero members, therefore AA = = øøA set can also have an infinite numbers of A set can also have an infinite numbers of

membersmembersExample: the set of natural numbersExample: the set of natural numbers

SubsetsSubsets If every member of If every member of AA is also a member of is also a member of BB, ,

then then AA is the subset of is the subset of BB, shown by, shown by We can also write it as We can also write it as BB is the superset of is the superset of AA, ,

shown by shown by is called inclusion or containmentis called inclusion or containment

A is a subset of XB is a subset of X

Proper SubsetsProper SubsetsHowever, if members of However, if members of AA is a subset, but is a subset, but

NOT equal to NOT equal to BB, then , then AA is called a proper is called a proper subset of subset of BB, written (or , written (or AA is a is a proper superset of proper superset of B, written B, written ). ).

can also be written as can also be written as can also be written ascan also be written as

Special SetsSpecial Sets All of these sets are represented using All of these sets are represented using

Blackboard bold typefaceBlackboard bold typeface denotes the set of all primesdenotes the set of all primes denotes the set of all natural numbersdenotes the set of all natural numbers denotes the set of all integers (positive, negative denotes the set of all integers (positive, negative

and zero)and zero) denotes the set of all rational numbersdenotes the set of all rational numbers denotes the set of all real numbersdenotes the set of all real numbers denotes the set of all complex numbersdenotes the set of all complex numbers

Although each of these sets have infinite size, Although each of these sets have infinite size, the order of the special sets is the order of the special sets is although the primes are used lessalthough the primes are used less

UnionUnion

Sets can be “added” togetherSets can be “added” togetherThe The unionunion of two of two setssets is the set of elements is the set of elements that are in at least one of the two that are in at least one of the two setssets. .

For For exampleexample, if A={1, 2, 3, 4} and B={2, 4, 6, 8}, if A={1, 2, 3, 4} and B={2, 4, 6, 8}

then A B = {1, 2, 3, 4, 6, 8}. then A B = {1, 2, 3, 4, 6, 8}.

““the union of A and B contains {1,2,3,4,6,8}the union of A and B contains {1,2,3,4,6,8}

IntersectionIntersection

The The intersectionintersection of two of two setssets is the elements they is the elements they have in common. have in common.

For For exampleexample, , if A={1, 2, 3, 4} and B={2, 4, 6, 8}, if A={1, 2, 3, 4} and B={2, 4, 6, 8},

then A B = {2, 4}. then A B = {2, 4}.

““the intersection between A and B contains the the intersection between A and B contains the numbers {2,4}numbers {2,4}

ComplementComplement

A` means everything except A. You can see A` means everything except A. You can see it from the diagram belowit from the diagram below

A` = X - A

ExampleExample

If there are 50 people in the class, 20 wears If there are 50 people in the class, 20 wears glasses and 24 wears watches and 15 wears glasses and 24 wears watches and 15 wears both, how many people do not wear anything?both, how many people do not wear anything?

X = {{G,W} – {B}}+{N}X = {{G,W} – {B}}+{N}G and W is a subset of XG and W is a subset of XX = everything within that rangeX = everything within that range = 50= 50(50) = (24) + (20) – (15) + N(50) = (24) + (20) – (15) + N = 29 + N N = 21= 29 + N N = 21

G WB

B

Challenge QuestionsChallenge QuestionsThere are 100 students in Year 10. In athletics day There are 100 students in Year 10. In athletics day

there are:there are: 52 people running 100 meters52 people running 100 meters 46 people running 200 meters46 people running 200 meters 33 people running 400 meters33 people running 400 meters 20 people running both 100 and 20020 people running both 100 and 200 16 people running both 200 and 40016 people running both 200 and 400 17 people running ONLY 20017 people running ONLY 200 20 people running ONLY 10020 people running ONLY 100

How many students are absent on that day?How many students are absent on that day?

The solutionThe solution 52 people running 100 meters52 people running 100 meters 46 people running 200 meters46 people running 200 meters 33 people running 400 meters33 people running 400 meters 20 people running both 100 and 20020 people running both 100 and 200 16 people running both 200 and 40016 people running both 200 and 400 17 people running ONLY 20017 people running ONLY 200 20 people running ONLY 10020 people running ONLY 100

More challenge questionsMore challenge questionsThere are 66 students and 19 teachers in ABC elementary There are 66 students and 19 teachers in ABC elementary

school. There are:school. There are: 36 boys36 boys 30 girls30 girls 19 teachers19 teachers 59 people wearing watches59 people wearing watches 56 people wearing glasses56 people wearing glasses The same number of boys and girls wearing glassesThe same number of boys and girls wearing glasses The number of teacher wearing glasses is 1 less than the The number of teacher wearing glasses is 1 less than the

number of boys wearing glassesnumber of boys wearing glasses 38 people wearing both38 people wearing both The number of girls wearing watches is 5 less than the The number of girls wearing watches is 5 less than the

number of boys wearing watchesnumber of boys wearing watches 20 girls wearing watches20 girls wearing watchesThere are 20 boys wearing only one objectThere are 20 boys wearing only one object There are 17 girls wearing only one objectThere are 17 girls wearing only one object

How many students do not wear neither watch or glasses?How many students do not wear neither watch or glasses?

Venn diagrams Venn diagrams Label/ shade these A’Label/ shade these A’

AB

Venn diagrams Venn diagrams Label/ shade these B’Label/ shade these B’

AB

Venn diagrams Venn diagrams Label/ shade these A BLabel/ shade these A B

AB

Venn diagrams Venn diagrams Label/ shade these A BLabel/ shade these A B

AB

Venn diagrams Venn diagrams Label/ shade these A’ BLabel/ shade these A’ B

AB

Venn diagrams Venn diagrams Label/ shade these A’ BLabel/ shade these A’ B

AB