1 sets cs/apma 202, spring 2005 rosen, section 1.6 aaron bloomfield

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SetsSetsCS/APMA 202, Spring 2005CS/APMA 202, Spring 2005

Rosen, section 1.6Rosen, section 1.6Aaron BloomfieldAaron Bloomfield

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Quick surveyQuick survey

How much set theory have you seen?How much set theory have you seen?

a)a) A lot – I’m an expertA lot – I’m an expert

b)b) A fair amountA fair amount

c)c) Not muchNot much

d)d) Set what?Set what?

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What is a set?What is a set?

A set is a group of “objects”A set is a group of “objects” People in a class: { Alice, Bob, Chris }People in a class: { Alice, Bob, Chris } Classes offered by a department: { CS 101, CS 202, … }Classes offered by a department: { CS 101, CS 202, … } Colors of a rainbow: { red, orange, yellow, green, blue, purple }Colors of a rainbow: { red, orange, yellow, green, blue, purple } States of matter { solid, liquid, gas, plasma }States of matter { solid, liquid, gas, plasma } States in the US: { Alabama, Alaska, Virginia, … }States in the US: { Alabama, Alaska, Virginia, … } Sets can contain non-related elements: { 3, a, red, Virginia }Sets can contain non-related elements: { 3, a, red, Virginia }

Although a set can contain (almost) anything, we will most Although a set can contain (almost) anything, we will most often use sets of numbersoften use sets of numbers

All positive numbers less than or equal to 5: {1, 2, 3, 4, 5}All positive numbers less than or equal to 5: {1, 2, 3, 4, 5} A few selected real numbers: { 2.1, A few selected real numbers: { 2.1, ππ, 0, -6.32, e }, 0, -6.32, e }

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Set properties 1Set properties 1

Order does not matterOrder does not matter We often write them in order because it is We often write them in order because it is

easier for humans to understand it that wayeasier for humans to understand it that way {1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1}{1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1}

Sets are notated with curly bracketsSets are notated with curly brackets

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Set properties 2Set properties 2

Sets do not have duplicate elementsSets do not have duplicate elements Consider the set of vowels in the alphabet. Consider the set of vowels in the alphabet.

It makes no sense to list them as {a, a, a, e, i, o, o, It makes no sense to list them as {a, a, a, e, i, o, o, o, o, o, u}o, o, o, u}What we really want is just {a, e, i, o, u}What we really want is just {a, e, i, o, u}

Consider the list of students in this classConsider the list of students in this classAgain, it does not make sense to list somebody Again, it does not make sense to list somebody twicetwice

Note that a list is like a set, but order does Note that a list is like a set, but order does matter and duplicate elements are allowedmatter and duplicate elements are allowed We won’t be studying lists much in this classWe won’t be studying lists much in this class

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Specifying a set 1Specifying a set 1

Sets are usually represented by a capital Sets are usually represented by a capital letter (A, B, S, etc.)letter (A, B, S, etc.)

Elements are usually represented by an Elements are usually represented by an italic lower-case letter (italic lower-case letter (aa, , xx, , yy, etc.), etc.)

Easiest way to specify a set is to list all the Easiest way to specify a set is to list all the elements: A = {1, 2, 3, 4, 5}elements: A = {1, 2, 3, 4, 5} Not always feasible for large or infinite setsNot always feasible for large or infinite sets

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Can use an ellipsis (…): B = {0, 1, 2, 3, …}Can use an ellipsis (…): B = {0, 1, 2, 3, …} Can cause confusion. Consider the set C = {3, 5, 7, Can cause confusion. Consider the set C = {3, 5, 7,

…}. What comes next?…}. What comes next? If the set is all odd integers greater than 2, it is 9If the set is all odd integers greater than 2, it is 9 If the set is all prime numbers greater than 2, it is 11If the set is all prime numbers greater than 2, it is 11

Can use set-builder notationCan use set-builder notation D = {D = {xx | | xx is prime and is prime and xx > 2} > 2} E = {E = {xx | | xx is odd and is odd and xx > 2} > 2} The vertical bar means “such that”The vertical bar means “such that” Thus, set D is read (in English) as: “all elements Thus, set D is read (in English) as: “all elements xx

such that such that xx is prime and is prime and xx is greater than 2” is greater than 2”

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A set is said to “contain” the various A set is said to “contain” the various “members” or “elements” that make up the “members” or “elements” that make up the setset If an element If an element aa is a member of (or an element is a member of (or an element

of) a set S, we use then notation of) a set S, we use then notation aa S S4 4 {1, 2, 3, 4} {1, 2, 3, 4}

If an element is not a member of (or an element If an element is not a member of (or an element of) a set S, we use the notation of) a set S, we use the notation aa S S

7 7 {1, 2, 3, 4} {1, 2, 3, 4}

Virginia Virginia {1, 2, 3, 4} {1, 2, 3, 4}

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Today’s demotivatorsToday’s demotivators

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Often used setsOften used sets

NN = {0, 1, 2, 3, …} is the set of natural numbers = {0, 1, 2, 3, …} is the set of natural numbersZZ = {…, -2, -1, 0, 1, 2, …} is the set of integers = {…, -2, -1, 0, 1, 2, …} is the set of integersZZ++ = {1, 2, 3, …} is the set of positive integers = {1, 2, 3, …} is the set of positive integers (a.k.a whole numbers)(a.k.a whole numbers) Note that people disagree on the exact definitions of Note that people disagree on the exact definitions of

whole numbers and natural numberswhole numbers and natural numbers

QQ = { = {pp//qq | | pp ZZ, , qq ZZ, , qq ≠ 0} is the set of ≠ 0} is the set of rational numbersrational numbers Any number that can be expressed as a fraction of Any number that can be expressed as a fraction of

two integers (where the bottom one is not zero)two integers (where the bottom one is not zero)

RR is the set of real numbers is the set of real numbers

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The universal set 1The universal set 1

UU is the universal set – the set of all of is the universal set – the set of all of elements (or the “universe”) from which elements (or the “universe”) from which given any set is drawngiven any set is drawn For the set {-2, 0.4, 2}, For the set {-2, 0.4, 2}, UU would be the real would be the real

numbersnumbers For the set {0, 1, 2}, For the set {0, 1, 2}, UU could be the natural could be the natural

numbers (zero and up), the integers, the numbers (zero and up), the integers, the rational numbers, or the real numbers, rational numbers, or the real numbers, depending on the contextdepending on the context

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The universal set 2The universal set 2

For the set of the students in this class, For the set of the students in this class, UU would be all the students in the University (or would be all the students in the University (or perhaps all the people in the world)perhaps all the people in the world)

For the set of the vowels of the alphabet, For the set of the vowels of the alphabet, UU would be all the letters of the alphabetwould be all the letters of the alphabet

To differentiate To differentiate UU from U (which is a set from U (which is a set operation), the universal set is written in a operation), the universal set is written in a different font (and in bold and italics)different font (and in bold and italics)

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Venn diagramsVenn diagrams

Represents sets graphicallyRepresents sets graphically The box represents the universal setThe box represents the universal set Circles represent the set(s)Circles represent the set(s)

Consider set S, which is Consider set S, which is the set of all vowels in thethe set of all vowels in thealphabetalphabet

The individual elements The individual elements are usually not written are usually not written in a Venn diagramin a Venn diagram

a e i

o u

b c d f

g h j

k l m

n p q

r s t

v w x

y z

US

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Sets of setsSets of sets

Sets can contain other setsSets can contain other sets S = { {1}, {2}, {3} }S = { {1}, {2}, {3} } T = { {1}, {{2}}, {{{3}}} }T = { {1}, {{2}}, {{{3}}} } V = { {{1}, {{2}}}, {{{3}}}, { {1}, {{2}}, {{{3}}} } }V = { {{1}, {{2}}}, {{{3}}}, { {1}, {{2}}, {{{3}}} } }

V has only 3 elements!V has only 3 elements!

Note that 1 ≠ {1} ≠ {{1}} ≠ {{{1}}}Note that 1 ≠ {1} ≠ {{1}} ≠ {{{1}}} They are all differentThey are all different

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The empty set 1The empty set 1

If a set has zero elements, it is called the If a set has zero elements, it is called the empty (or null) setempty (or null) set Written using the symbol Written using the symbol Thus, Thus, = { } = { } VERY IMPORTANT VERY IMPORTANT If you get confused about the empty set in a If you get confused about the empty set in a

problem, try replacing problem, try replacing by { } by { }

As the empty set is a set, it can be a As the empty set is a set, it can be a element of other setselement of other sets { { , 1, 2, 3, x } is a valid set, 1, 2, 3, x } is a valid set

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The empty set 1The empty set 1

Note that Note that ≠ { ≠ { } } The first is a set of zero elementsThe first is a set of zero elements The second is a set of 1 element (that one The second is a set of 1 element (that one

element being the empty set)element being the empty set)

Replace Replace by { }, and you get: { } ≠ { { } } by { }, and you get: { } ≠ { { } }It’s easier to see that they are not equal that wayIt’s easier to see that they are not equal that way

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Empty set: an exampleEmpty set: an example

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Set equalitySet equality

Two sets are equal if they have the same Two sets are equal if they have the same elementselements {1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}{1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}

Remember that order does not matter!Remember that order does not matter! {1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2, 1}{1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2, 1}

Remember that duplicate elements do not matter!Remember that duplicate elements do not matter!

Two sets are not equal if they do not have Two sets are not equal if they do not have the same elementsthe same elements {1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}{1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}

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Subsets 1Subsets 1

If all the elements of a set S are also elements of If all the elements of a set S are also elements of a set T, then S is a subset of Ta set T, then S is a subset of T For example, if S = {2, 4, 6} and T = {1, 2, 3, 4, 5, 6, For example, if S = {2, 4, 6} and T = {1, 2, 3, 4, 5, 6,

7}, then S is a subset of T7}, then S is a subset of T This is specified by S This is specified by S T T

Or by {2, 4, 6} Or by {2, 4, 6} {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}

If S is not a subset of T, it is written as such: If S is not a subset of T, it is written as such: S S T T For example, {1, 2, 8} For example, {1, 2, 8} {1, 2, 3, 4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}

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Subsets 2Subsets 2

Note that any set is a subset of itself!Note that any set is a subset of itself! Given set S = {2, 4, 6}, since all the elements Given set S = {2, 4, 6}, since all the elements

of S are elements of S, S is a subset of itselfof S are elements of S, S is a subset of itself This is kind of like saying 5 is less than or This is kind of like saying 5 is less than or

equal to 5equal to 5 Thus, for any set S, S Thus, for any set S, S S S

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Subsets 3Subsets 3

The empty set is a subset of The empty set is a subset of allall sets (including sets (including itself!)itself!) Recall that all sets are subsets of themselvesRecall that all sets are subsets of themselves

AllAll sets are subsets of the universal set sets are subsets of the universal setThe textbook has this gem to define a subset:The textbook has this gem to define a subset: xx ( ( xxA A xxB )B ) English translation: for all possible values of x, English translation: for all possible values of x,

(meaning for all possible elements of a set), if x is an (meaning for all possible elements of a set), if x is an element of A, then x is an element of Belement of A, then x is an element of B

This type of notation will be gone over laterThis type of notation will be gone over later

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If S is a subset of T, and S is not equal to T, If S is a subset of T, and S is not equal to T, then S is a proper subset of Tthen S is a proper subset of T Let T = {0, 1, 2, 3, 4, 5}Let T = {0, 1, 2, 3, 4, 5} If S = {1, 2, 3}, S is not equal to T, and S is a If S = {1, 2, 3}, S is not equal to T, and S is a

subset of Tsubset of T A proper subset is written as S A proper subset is written as S T T Let R = {0, 1, 2, 3, 4, 5}. R is equal to T, and Let R = {0, 1, 2, 3, 4, 5}. R is equal to T, and

thus is a subset (but not a proper subset) or Tthus is a subset (but not a proper subset) or TCan be written as: R Can be written as: R T and R T and R T (or just R = T) T (or just R = T)

Let Q = {4, 5, 6}. Q is neither a subset or T nor Let Q = {4, 5, 6}. Q is neither a subset or T nor a proper subset of Ta proper subset of T

Proper Subsets 1Proper Subsets 1

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Proper Subsets 2Proper Subsets 2

The difference between “subset” and The difference between “subset” and “proper subset” is like the difference “proper subset” is like the difference between “less than or equal to” and “less between “less than or equal to” and “less than” for numbersthan” for numbers

The empty set is a proper subset of all The empty set is a proper subset of all sets other than the empty set (as it is sets other than the empty set (as it is equal to the empty set)equal to the empty set)

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Proper subsets: Venn diagramProper subsets: Venn diagram

U

S

R

S R

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I understand the difference between I understand the difference between regular subsets and proper subsetsregular subsets and proper subsets

a)a) Very wellVery well

b)b) With some review, I’ll be goodWith some review, I’ll be good

c)c) Not reallyNot really

d)d) Not at allNot at all

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A bit of humor…A bit of humor…

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Set cardinalitySet cardinality

The cardinality of a set is the number of elements in a The cardinality of a set is the number of elements in a setset Written as |A|Written as |A|

ExamplesExamples Let R = {1, 2, 3, 4, 5}. Then |R| = 5Let R = {1, 2, 3, 4, 5}. Then |R| = 5 ||| = 0| = 0 Let S = {Let S = {, {a}, {b}, {a, b}}. Then |S| = 4, {a}, {b}, {a, b}}. Then |S| = 4

This is the same notation used for vector length in This is the same notation used for vector length in geometrygeometry

A set with one element is sometimes called a A set with one element is sometimes called a singleton setsingleton set

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Power sets 1Power sets 1

Given the set S = {0, 1}. What are all the Given the set S = {0, 1}. What are all the possible subsets of S?possible subsets of S? They are: They are: (as it is a subset of all sets), {0}, (as it is a subset of all sets), {0},

{1}, and {0, 1}{1}, and {0, 1} The power set of S (written as P(S)) is the set The power set of S (written as P(S)) is the set

of all the subsets of Sof all the subsets of S P(S) = { P(S) = { , {0}, {1}, {0,1} }, {0}, {1}, {0,1} }

Note that |S| = 2 and |P(S)| = 4Note that |S| = 2 and |P(S)| = 4

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Power sets 2Power sets 2

Let T = {0, 1, 2}. The P(T) = { Let T = {0, 1, 2}. The P(T) = { , {0}, {1}, , {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2} }{2}, {0,1}, {0,2}, {1,2}, {0,1,2} }

Note that |T| = 3 and |P(T)| = 8Note that |T| = 3 and |P(T)| = 8

P(P() = { ) = { } }Note that |Note that || = 0 and |P(| = 0 and |P()| = 1)| = 1

If a set has If a set has nn elements, then the power set elements, then the power set will have 2will have 2nn elements elements

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TuplesTuples

In 2-dimensional space, it is a (In 2-dimensional space, it is a (xx, , yy) pair of ) pair of numbers to specify a locationnumbers to specify a locationIn 3-dimensional (1,2,3) is not the same as In 3-dimensional (1,2,3) is not the same as (3,2,1) – space, it is a ((3,2,1) – space, it is a (xx, , yy, , zz) triple of numbers) triple of numbersIn In nn-dimensional space, it is a -dimensional space, it is a nn-tuple of numbers-tuple of numbers Two-dimensional space uses Two-dimensional space uses

pairs, or 2-tuplespairs, or 2-tuples Three-dimensional space uses Three-dimensional space uses

triples, or 3-tuplestriples, or 3-tuples

Note that these tuples are Note that these tuples are orderedordered, unlike sets, unlike sets the the xx value has to come first value has to come first

+x

+y

(2,3)

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Cartesian products 1Cartesian products 1

A Cartesian product is a set of all ordered 2-A Cartesian product is a set of all ordered 2-tuples where each “part” is from a given settuples where each “part” is from a given set Denoted by A x B, and uses parenthesis (not curly Denoted by A x B, and uses parenthesis (not curly

brackets)brackets) For example, 2-D Cartesian coordinates are the set of For example, 2-D Cartesian coordinates are the set of

all ordered pairs all ordered pairs ZZ x x ZZRecall Recall ZZ is the set of all integers is the set of all integers

This is all the possible coordinates in 2-D spaceThis is all the possible coordinates in 2-D space Example: Given A = { a, b } and B = { 0, 1 }, what is Example: Given A = { a, b } and B = { 0, 1 }, what is

their Cartiesian product?their Cartiesian product?C = A x B = { (a,0), (a,1), (b,0), (b,1) }C = A x B = { (a,0), (a,1), (b,0), (b,1) }

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Note that Cartesian products have only 2 Note that Cartesian products have only 2 parts in these examples (later examples parts in these examples (later examples have more parts)have more parts)

Formal definition of a Cartesian product:Formal definition of a Cartesian product: A x B = { (A x B = { (aa,,bb) | ) | aa A and A and bb B } B }

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All the possible grades in this class will be a All the possible grades in this class will be a Cartesian product of the set S of all the students Cartesian product of the set S of all the students in this class and the set G of all possible gradesin this class and the set G of all possible grades Let S = { Alice, Bob, Chris } and G = { A, B, C }Let S = { Alice, Bob, Chris } and G = { A, B, C } D = { (Alice, A), (Alice, B), (Alice, C), (Bob, A), (Bob, D = { (Alice, A), (Alice, B), (Alice, C), (Bob, A), (Bob,

B), (Bob, C), (Chris, A), (Chris, B), (Chris, C) }B), (Bob, C), (Chris, A), (Chris, B), (Chris, C) } The final grades will be a subset of this: { (Alice, C), The final grades will be a subset of this: { (Alice, C),

(Bob, B), (Chris, A) }(Bob, B), (Chris, A) }Such a subset of a Cartesian product is called a relation Such a subset of a Cartesian product is called a relation (more on this later in the course)(more on this later in the course)

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There can be Cartesian products on more There can be Cartesian products on more than two setsthan two sets

A 3-D coordinate is an element from the A 3-D coordinate is an element from the Cartesian product of Cartesian product of ZZ x x ZZ x x ZZ

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Using set notation with quantifiersUsing set notation with quantifiers

Ignore this part of the section (pages 84-Ignore this part of the section (pages 84-85) for now85) for now We’ll discuss it after we look at quantifiers We’ll discuss it after we look at quantifiers

(sections 1.3-1.4)(sections 1.3-1.4)

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I felt I understood the material in this I felt I understood the material in this slide set…slide set…

a)a) Very wellVery well

b)b) With some review, I’ll be goodWith some review, I’ll be good

c)c) Not reallyNot really

d)d) Not at allNot at all

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The pace of the lecture for this The pace of the lecture for this slide set was…slide set was…

a)a) FastFast

b)b) About rightAbout right

c)c) A little slowA little slow

d)d) Too slowToo slow

1 sets cs/apma 202, spring 2005 rosen, section 1.6 aaron bloomfield

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