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Sets and Counting

Sets and Set Operations

Cardinality

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Asetis a collection of items called theelements of the set.

Ex.A= {w, a, r, d}

Sets are often named

with capital letters.

Order of elements doesnt

matter, no dplicates.

w is an element of setAis written

m is not an element of setAis written

Sets and Elements

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!wo setsAandBare equal, writtenA=B, if

and only if they ha"e exactly the same

elements.

Ex.A= {w, a, r, d}#B= {d, r, a, w}

E"ery element inAis inB ande"ery element inBis inA.

Set E$ality

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%f e"ery element of a setAis also an element of a set

BthenAis asubsetofB, and is written AB

Ex.A= {r, d}#B= {r, a, w, d, e, t}

E"ery element inAis also inB

S&sets

A B means thatAis aproper subsetofB'

AB, &tAB

!he nm&er of ' s&sets in a set = ,(n

proper s&sets = .( )n

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!he set that contains no elements is called

the empty setand is denoted { } or

*ote' !he empty set is a s&set of e"ery set

Empty Set

Afinite sethas finitely many elements. An

infinite setdoes not ha"e finitely many elements.

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Set of Otcomes

%f yo flip a penny the possi&le otcomes

are heads +h and tails +t. !he set of

otcomes can &e represented &y'

S = {h, t}

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A rle descri&es the definite property

+properties an o&-ectxmst satisfy to

&e part of the set.

Ex.B= {x xis an e"en inte/er}

0ead' 1xsch thatxis an e"en inte/er2

Set34ilder *otation

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5enn 6ia/rams

A

7 "isal representation of setsSets are

represented

&y circles

B

A

Ex. !his 5ennrepresents &oth

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Set Union' 8etAandB&e sets. !he nion ofAand

B, written is the set of all elements that &elon/

to eitherAorB.

Set Operations

BA

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Set Intersection' 8etAandB&e sets. !heintersection ofAandB, written is the

set of all elements that are common toAand

B.

Set Operations

%f then setsAandBare said to &e

disjoint.

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Complement of a Set: %fAis a s&set of S, then

the set of all elements in Sthat are not inAis

called the complement ofA, written

Set Operations

A

.A

S

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5enn 6ia/rams 7 9 Sets

S

A B

C

A

C

B

S

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Ex. :i"en the sets'

Com&ine the sets

O"erlap of the sets

Elements in Snot

inA.

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!he set of all elements of interest in a

particlar discssion is called the universalsetand is denoted U or .

!he ;ni"ersal Set

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Cartesian

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Cartesian

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Cardinality

!he cardinality of a finite setAis the nm&er

of elements the set and is denoted n+A.

Ex. :i"enA= {r, a, t, s}

n+A =

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Cardinality of a ;nion

%fAandBare finite sets,

Ex. :i"en

Since the nion doesnt

cont aand htwice

O"erlap iss&tracted

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Cardinality of a Complement

%f Sis a finite ni"ersal set andAis as&set of S, then

Ex. :i"en

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Cardinality of a Cartesian

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5enn 6ia/ram

SA B

7 nm&er of elements

Ex. :i"en

;se this information to fill in the 5enn dia/ram a&o"e.

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Sr"ey%n a sr"ey of ) people at a carni"al'

> li?e cotton candy

9 li?e popcorn

>@ li?e lemonade

)@ li?e lemonade and popcorn

) li?e cotton candy and lemonade

)( li?e cotton candy and popcorn@ li?e all three

ow many people dont li?e lemonade, popcorn or cotton

candyB

ow many people only li?e popcornB