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Sets

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What is a set? A set is a group of “objects” People in a class: { Alice, Bob, Chris } Classes offered by a department: { CS 101, CS 202, … } Colors of a rainbow: { red, orange, yellow, green, blue, purple } States of matter { solid, liquid, gas, plasma } States in the US: { Alabama, Alaska, Virginia, … } Sets can contain non-related elements: { 3, a, red, Virginia } Although a set can contain (almost) anything, we will most often use sets of numbers All positive numbers less than or equal to 5: {1, 2, 3, 4, 5} A few selected real numbers: { 2.1, π, 0, -6.32, e }

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Set properties 1 Order does not matter We often write them in order because it is easier for humans to understand it that way {1, 2, 3, 4, 5} is equivalent to {3, 5, 2, 4, 1} Sets are notated with curly brackets

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Set properties 2 Sets do not have duplicate elements Consider the set of vowels in the alphabet. It makes no sense to list them as {a, a, a, e, i, o, o, o, o, o, u} What we really want is just {a, e, i, o, u} Consider the list of students in this class Again, it does not make sense to list somebody twice Note that a list is like a set, but order does matter and duplicate elements are allowed We won’t be studying lists much in this class

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Specifying a set 1 Sets are usually represented by a capital letter (A, B, S, etc.) Elements are usually represented by an italic lower-case letter (a, x, y, etc.) Easiest way to specify a set is to list all the elements: A = {1, 2, 3, 4, 5} Not always feasible for large or infinite sets

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Specifying a set 2 Can use an ellipsis (…): B = {0, 1, 2, 3, …} Can cause confusion. Consider the set C = {3, 5, 7, …}. What comes next? If the set is all odd integers greater than 2, it is 9 If the set is all prime numbers greater than 2, it is 11 Can use set-builder notation D = {x | x is prime and x > 2} E = {x | x is odd and x > 2} The vertical bar means “such that” Thus, set D is read (in English) as: “all elements x such that x is prime and x is greater than 2”

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Specifying a set 3 A set is said to “contain” the various “members” or “elements” that make up the set If an element a is a member of (or an element of) a set S, we use then notation a  S 4  {1, 2, 3, 4} If an element is not a member of (or an element of) a set S, we use the notation a  S 7  {1, 2, 3, 4} Virginia  {1, 2, 3, 4}

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Often used sets N = {0, 1, 2, 3, …} is the set of natural numbers Z = {…, -2, -1, 0, 1, 2, …} is the set of integers Z+ = {1, 2, 3, …} is the set of positive integers (a.k.a whole numbers) Note that people disagree on the exact definitions of whole numbers and natural numbers Q = {p/q | p  Z, q  Z, q ≠ 0} is the set of rational numbers Any number that can be expressed as a fraction of two integers (where the bottom one is not zero) R is the set of real numbers

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The universal set 1 U is the universal set – the set of all of elements (or the “universe”) from which given any set is drawn For the set {-2, 0.4, 2}, U would be the real numbers For the set {0, 1, 2}, U could be the natural numbers (zero and up), the integers, the rational numbers, or the real numbers, depending on the context

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The universal set 2 For the set of the students in this class, U would be all the students in the University (or perhaps all the people in the world) For the set of the vowels of the alphabet, U would be all the letters of the alphabet To differentiate U from U (which is a set operation), the universal set is written in a different font (and in bold and italics)

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Venn diagrams Represents sets graphically The box represents the universal set Circles represent the set(s) Consider set S, which is the set of all vowels in the alphabet The individual elements are usually not written in a Venn diagram

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Sets of sets Sets can contain other sets S = { {1}, {2}, {3} } T = { {1}, {{2}}, {{{3}}} } V = { {{1}, {{2}}}, {{{3}}}, { {1}, {{2}}, {{{3}}} } } V has only 3 elements! Note that 1 ≠ {1} ≠ {{1}} ≠ {{{1}}} They are all different

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The empty set 1 If a set has zero elements, it is called the empty (or null) set Written using the symbol  Thus,  = { }  VERY IMPORTANT If you get confused about the empty set in a problem, try replacing  by { } As the empty set is a set, it can be a element of other sets { , 1, 2, 3, x } is a valid set

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The empty set 1 Note that  ≠ {  } The first is a set of zero elements The second is a set of 1 element (that one element being the empty set) Replace  by { }, and you get: { } ≠ { { } } It’s easier to see that they are not equal that way

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Set equality Two sets are equal if they have the same elements {1, 2, 3, 4, 5} = {5, 4, 3, 2, 1} Remember that order does not matter! {1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2, 1} Remember that duplicate elements do not matter! Two sets are not equal if they do not have the same elements {1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}

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Subsets 1 If all the elements of a set S are also elements of a set T, then S is a subset of T For example, if S = {2, 4, 6} and T = {1, 2, 3, 4, 5, 6, 7}, then S is a subset of T This is specified by S  T Or by {2, 4, 6}  {1, 2, 3, 4, 5, 6, 7} If S is not a subset of T, it is written as such: S  T For example, {1, 2, 8}  {1, 2, 3, 4, 5, 6, 7}

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Subsets 2 Note that any set is a subset of itself! Given set S = {2, 4, 6}, since all the elements of S are elements of S, S is a subset of itself This is kind of like saying 5 is less than or equal to 5 Thus, for any set S, S  S

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Subsets 3 The empty set is a subset of all sets (including itself!) Recall that all sets are subsets of themselves All sets are subsets of the universal set A horrible way to define a subset: x ( xA  xB ) English translation: for all possible values of x, (meaning for all possible elements of a set), if x is an element of A, then x is an element of B This type of notation will be gone over later

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Proper Subsets 1 If S is a subset of T, and S is not equal to T, then S is a proper subset of T Let T = {0, 1, 2, 3, 4, 5} If S = {1, 2, 3}, S is not equal to T, and S is a subset of T A proper subset is written as S  T Let R = {0, 1, 2, 3, 4, 5}. R is equal to T, and thus is a subset (but not a proper subset) or T Can be written as: R  T and R  T (or just R = T) Let Q = {4, 5, 6}. Q is neither a subset or T nor a proper subset of T

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Proper Subsets 2 The difference between “subset” and “proper subset” is like the difference between “less than or equal to” and “less than” for numbers The empty set is a proper subset of all sets other than the empty set (as it is equal to the empty set)

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Proper subsets: Venn diagram

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Set cardinality The cardinality of a set is the number of elements in a set Written as |A| Examples Let R = {1, 2, 3, 4, 5}. Then |R| = 5 || = 0 Let S = {, {a}, {b}, {a, b}}. Then |S| = 4 This is the same notation used for vector length in geometry A set with one element is sometimes called a singleton set

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Power sets 1 Given the set S = {0, 1}. What are all the possible subsets of S? They are:  (as it is a subset of all sets), {0}, {1}, and {0, 1} The power set of S (written as P(S)) is the set of all the subsets of S P(S) = { , {0}, {1}, {0,1} } Note that |S| = 2 and |P(S)| = 4

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Power sets 2 Let T = {0, 1, 2}. The P(T) = { , {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2} } Note that |T| = 3 and |P(T)| = 8 P() = {  } Note that || = 0 and |P()| = 1 If a set has n elements, then the power set will have 2n elements

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Tuples In 2-dimensional space, it is a (x, y) pair of numbers to specify a location In 3-dimensional (1,2,3) is not the same as (3,2,1) – space, it is a (x, y, z) triple of numbers In n-dimensional space, it is a n-tuple of numbers Two-dimensional space uses pairs, or 2-tuples Three-dimensional space uses triples, or 3-tuples Note that these tuples are ordered, unlike sets the x value has to come first

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Cartesian products 1 A Cartesian product is a set of all ordered 2-tuples where each “part” is from a given set Denoted by A x B, and uses parenthesis (not curly brackets) For example, 2-D Cartesian coordinates are the set of all ordered pairs Z x Z Recall Z is the set of all integers This is all the possible coordinates in 2-D space Example: Given A = { a, b } and B = { 0, 1 }, what is their Cartiesian product? C = A x B = { (a,0), (a,1), (b,0), (b,1) }

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Cartesian products 2 Note that Cartesian products have only 2 parts in these examples (later examples have more parts) Formal definition of a Cartesian product: A x B = { (a,b) | a  A and b  B }

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Cartesian products 3 All the possible grades in this class will be a Cartesian product of the set S of all the students in this class and the set G of all possible grades Let S = { Alice, Bob, Chris } and G = { A, B, C } D = { (Alice, A), (Alice, B), (Alice, C), (Bob, A), (Bob, B), (Bob, C), (Chris, A), (Chris, B), (Chris, C) } The final grades will be a subset of this: { (Alice, C), (Bob, B), (Chris, A) } Such a subset of a Cartesian product is called a relation (more on this later in the course)

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Cartesian products 4 There can be Cartesian products on more than two sets A 3-D coordinate is an element from the Cartesian product of Z x Z x Z

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Set Operations

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Set operations: Union

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Set operations: Union Formal definition for the union of two sets: A U B = { x | x  A or x  B } Further examples {1, 2, 3} U {3, 4, 5} = {1, 2, 3, 4, 5} {New York, Washington} U {3, 4} = {New York, Washington, 3, 4} {1, 2} U  = {1, 2}

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Set operations: Union Properties of the union operation A U  = A Identity law A U U = U Domination law A U A = A Idempotent law A U B = B U A Commutative law A U (B U C) = (A U B) U C Associative law

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Set operations: Intersection

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Set operations: Intersection Formal definition for the intersection of two sets: A ∩ B = { x | x  A and x  B } Further examples {1, 2, 3} ∩ {3, 4, 5} = {3} {New York, Washington} ∩ {3, 4} =  No elements in common {1, 2} ∩  =  Any set intersection with the empty set yields the empty set

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Set operations: Intersection Properties of the intersection operation A ∩ U = A Identity law A ∩  =  Domination law A ∩ A = A Idempotent law A ∩ B = B ∩ A Commutative law A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative law

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Disjoint sets 1

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Disjoint sets 2

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Disjoint sets 3 Formal definition for disjoint sets: two sets are disjoint if their intersection is the empty set Further examples {1, 2, 3} and {3, 4, 5} are not disjoint {New York, Washington} and {3, 4} are disjoint {1, 2} and  are disjoint Their intersection is the empty set  and  are disjoint! Their intersection is the empty set

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Set operations: Difference

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Set operations: Difference Formal definition for the difference of two sets: A - B = { x | x  A and x  B } A - B = A ∩ B  Important! Further examples {1, 2, 3} - {3, 4, 5} = {1, 2} {New York, Washington} - {3, 4} = {New York, Washington} {1, 2} -  = {1, 2} The difference of any set S with the empty set will be the set S

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Set operations: Symmetric Difference A symmetric difference of the sets contains all the elements in either set but NOT both Formal definition for the symmetric difference of two sets: A  B = { x | (x  A or x  B) and x  A ∩ B} A  B = (A U B) – (A ∩ B)  Important! Further examples {1, 2, 3}  {3, 4, 5} = {1, 2, 4, 5} {New York, Washington}  {3, 4} = {New York, Washington, 3, 4} {1, 2}   = {1, 2} The symmetric difference of any set S with the empty set will be the set S

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Complement sets A complement of a set is all the elements that are NOT in the set Formal definition for the complement of a set: A = { x | x  A } Further examples (assuming U = Z) {1, 2, 3} = { …, -2, -1, 0, 4, 5, 6, … }