lecture set i _ sets

8/7/2019 Lecture Set I _ Sets

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Copyright Andrew Katumba,2011

CMP 2201 Lecture Set I:Sets

Andrew Katumba


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L4 2

AgendaSets Curly brace notation { }

Cardinality | |

Element containment Subset containment

Empty set{ } =

Power set P(S ) = 2S

N-tuples ( ) and Cartesian productv


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L4 3

Section 1.4: Sets

DEF: A setis a collection of elements.

This is another example where mathematics

must start at the level of intuition. Sets arethe basic data structure out of which mostmathematical theories are built. For manyyears mathematicians hoped that sets could

be defined directly from logic, thus giving afull-proof foundation to Mathematics, whencompared to other sciences. Effort failed!


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L4 4

SetsCurly braces { and } are used to denote

sets.

Java note: In Java curly braces denote arrays,

a data-structure with inherent ordering.Mathematical sets are unordered so differentfrom Java arrays. Java arrays require that allelements be of the same type. Mathematical

sets don

t require this, however. EG: { 11, 12, 13 }

{ , , }

{ , , , 11, Leo }


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L4 5

Sets

A set is defined only by the elementswhich it contains. Thus repeating an

element, or changing the ordering ofelements in the description of the set,does not change the set itself:

{ 11, 11, 11, 12, 13 } = { 11, 12, 13 } { , , } = { , , }


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L4 6

Standard Numerical SetsThe natural numbers:N= { 0, 1, 2, 3, 4, }

The integers:

Z = { -3, -2, -1, 0, 1, 2, 3, }The positive integers:Z+ = {1, 2, 3, 4, 5, }

The real numbers: R --contains any decimal

number of arbitrary precisionThe rational numbers: Q --these are decimalnumbers whose decimal expansion repeats

Q: Give examples of numbers in Rbut notQ.


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Standard Numerical Sets

A: num erirrati nalanr,,,2 e


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-NotationThe Greek letter (epsilon) is used to denote

that an object is an element of a set. Whencrossed out denotes that the object isnot an element.

EG: 3 S reads:

3 is an element of the setS.

Q: Which of the following are true:

1. 3 R2. -3 N

3. -3 R

4. 0 Z+

5. x xR x2

=-5


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-Notation

A: 1, 3 and 4

1. 3 R. True: 3 is a real number.

2. -3 N. False: natural numbers dontcontain negatives.

3. -3 R. True: -3 is a real number.

4. 0 Z+. True: 0 isnt positive.

5. x xR x2=-5 . False: square of areal number is non-neg., so cant be -5.


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L4 10

-NotationDEF: A setS is said to be a subsetof the set T

iff every element ofS is also an element ofT. This situation is denoted by

S TA synonym ofsubset is contained by.

Definitions are often just a means ofestablishing a logical equivalence which aids

in notation. The definition above says that:S T x (xS ) p (xT)

We already had all the necessary concepts, butthe notation saves work.


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L4 11

-NotationWhen is used instead of, proper

containment is meant. A subsetSofTissaid to be a propersubsetifS is not equal

to T. Notationally:

S T ST x (x S xT)

Q: What algebraic symbol is reminiscent of?


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L4 12

-Notation

A: is to , as < is to e.


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L4 13

The Empty Set

The empty set is the set containing noelements. This set is also called the

null set and is denoted by: {}


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L4 14

SubsetExamples

Q: Which of the following are true:

1. N R

2. Z N

3. -3 R

4. {1,2} Z+

5.

6.


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L4 15

SubsetExamples

A: 1, 4 and 51. N R. All natural numbers are real.

2. Z N. Negative numbers aren

t natural.3. -3 R. Nonsensical. -3 is not a subset but an

element! (This could have made sense if weviewed -3 as a setwhich in principle is thecase in this case the proposition is false).

4. {1,2} Z+

. This actually makes sense. The set{1,2} is an object in its own right, so could bean element of some set; however, {1,2} is nota number, therefore is not an element ofZ.

5. . Any set contains itself.

6. . No set can contain itself properly.


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L4 16

Cardinality

The cardinalityof a set is the number ofdistinct elements in the set. |S|

denotes the cardinality ofS.Q: Compute each cardinality.

1. |{1, -13, 4, -13, 1}|

2. |{3, {1,2,3,4}, }|

3. |{}|

4. |{{}, {{}}, {{{}}} }|


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Cardinality

Hint: After eliminating the redundancies justlook at the number of top level commas andadd 1 (except for the empty set).

A:1. |{1, -13, 4, -13, 1}| = |{1, -13, 4}| = 3

2. |{3, {1,2,3,4}, }| = 3. To see this, setS={1,2,3,4}. Compute the cardinality of{3,S, }

3. |{}| = || = 0

4. |{{}, {{}}, {{{}}} }|= |{, {}, {{}}| = 3


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L4 18

Cardinality

DEF: The setS is said to be finite if itscardinality is a nonnegative integer.Otherwise, S is said to be infinite.

EG: N, Z, Z+, R, Q are each infinite.

Note: Well see later that not all infinities are

the same. In fact, R will end up having abigger infinity-type than N, but surprisingly,Nhas same infinity-type as Z, Z+, and Q.


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Power Set

DEF: The powersetof S is the set of allsubsets of S.

Denote the power set by P (S ) or by 2s .

The latter weird notation comes from thefollowing.

Lemma: | 2s | = 2|s|


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L4 20

Power SetExampleTo understand the previous fact consider

S = {1,2,3}

Enumerate all the subsets of S :

0-element sets: {} 1

1-element sets: {1}, {2}, {3} +3

2-element sets: {1,2}, {1,3}, {2,3} +3

3-element sets: {1,2,3} +1Therefore: | 2s | = 8 = 23 = 2|s|


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L4 21

Ordered n-tuplesNotationally, n-tuples look like sets except that

curly braces are replaced by parentheses: ( 11, 12 ) a 2-tuple aka orderedpair

( , , )

a 3-tuple ( , , , 11, Leo ) a 5-tuple

Java: n -tuples are similar to Java arrays {},

except that type-mixing isnt allowed in Java.


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L4 22

Ordered n-tuples

As opposed to sets, repetition andordering do matter with n-tuples.

(11, 11, 11, 12, 13){

( 11, 12, 13 ) ( , , ) { ( , , )


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L4 23

Cartesian ProductThe most famous example of 2-tuples are

points in the Cartesian plane R2. Hereordered pairs (x,y) of elements ofR describethe coordinates of each point. We can thinkof the first coordinate as the value on the x-axis and the second coordinate as the valueon the y-axis.

DEF: The Cartesianproductof two sets Aand B

denoted by A vB

is the set of all

ordered pairs (a, b) where aAand bB .

Q: Describe R2 as the Cartesian product of twosets.


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L4 24

Cartesian ProductA: R2 = RvR. I.e., the Cartesian plane is

formed by taking the Cartesian productof the x-axis with the y-axis.

One can generalize the Cartesian productto several sets simultaneously.

Q: IfA= {1,2}, B = {3,4}, C = {5,6,7}

what is AvBvC ?


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Cartesian Product A: A ={1,2}, B = {3,4}, C = {5,6,7}

A vBvC =

{ (1,3,5), (1,3,6), (1,3,7),

(1,4,5), (1,4,6), (1,4,7),(2,3,5), (2,3,6), (2,3,7),

(2,4,5), (2,4,6), (2,4,7) }

Lemma: The cardinality of the Cartesianproduct is the product of the cardinalities:

| A1 vA2 v v An| = |A1||A2| |An|

Q: What does vSequal?


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Cartesian Product

A: From the lemma:

|vS | = |||S | = 0|S | = 0

There is only one set with no elements

the empty set therefore, vSmust bethe empty set.

One can also check this directly from thedefinition of the Cartesian product.


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Exercise

Prove the following:

IfA B and B C then A C .

lecture set i _ sets

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