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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
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