Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter1 Module #3: The Theory of Sets Rosen 5 th ed., §§

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter2 Introduction to Set Theory (§1.6) A set is another type of structure, representing an unordered collection of zero or more distinct objects.A set is another type of structure, representing an unordered collection of zero or more distinct objects. Sets are ubiquitous in computer software systems.Sets are ubiquitous in computer software systems. Probably all of mathematics can be defined in terms of some form of set theory.Probably all of mathematics can be defined in terms of some form of set theory. –Relations, functions, etc.

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter3 Intuition behind sets Almost anything you can do with individual objects, you can also do with sets of objects. E.g. (informally speaking), you canAlmost anything you can do with individual objects, you can also do with sets of objects. E.g. (informally speaking), you can –refer to them, compare them, combine them, … You can also do some things to a set that you probably cannot do to an individual: E.g., you canYou can also do some things to a set that you probably cannot do to an individual: E.g., you can –check whether one set is contained in another (?) –determine how many elements it has (?) –quantify over its elements (using it as u.d. for ,  )

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter4 Basic notations for sets For sets, we ’ ll use variables S, T, U, …For sets, we ’ ll use variables S, T, U, … We can denote a set S in writing by listing all of its elements in curly braces:We can denote a set S in writing by listing all of its elements in curly braces: –{a, b, c} is the set of whatever 3 objects are denoted by a, b, c. Set builder notation: For any proposition P(x) over any universe of discourse, {x|P(x)} is the set of all x such that P(x).Set builder notation: For any proposition P(x) over any universe of discourse, {x|P(x)} is the set of all x such that P(x).

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter5 Basic properties of sets Sets are inherently unordered:Sets are inherently unordered: –No matter what objects a, b, and c denote, {a, b, c} = {a, c, b} = {b, a, c} = … Multiple listings make no difference:Multiple listings make no difference: –{a, a, c, c, c, c}={a,c}.

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter6 Basic properties of sets There exists a different mathematical construct, called bag or multiset, where this assumption does not hold. Using square brackets, we haveThere exists a different mathematical construct, called bag or multiset, where this assumption does not hold. Using square brackets, we have –[a,a,c,c,c,c]=[a,c,a,c,c,c]  [a,a,a,c] Notation: if B is a bag then count B (e)=number of occurrences of e in BNotation: if B is a bag then count B (e)=number of occurrences of e in B

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter7 Definition of Set Equality Two sets are equal if and only if they contain exactly the same elements.Two sets are equal if and only if they contain exactly the same elements. It does not matter how the set is definedIt does not matter how the set is defined For example: {1, 2, 3, 4} = {x | x is an integer where x>0 and x 0 and 0 and x 0 and <25}

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter8 Infinite Sets Sets may be infinite (i.e., not finite, without end, unending).Sets may be infinite (i.e., not finite, without end, unending). Symbols for some special infinite sets: N = {0, 1, 2, …} The Natural numbers. Z = {…, -2, -1, 0, 1, 2, …} The integers. R = The “ Real ” numbers, such as …Symbols for some special infinite sets: N = {0, 1, 2, …} The Natural numbers. Z = {…, -2, -1, 0, 1, 2, …} The integers. R = The “ Real ” numbers, such as … “ Blackboard Bold ” or double-struck font ( ℕ, ℤ, ℝ ) is also often used for these special number sets.“ Blackboard Bold ” or double-struck font ( ℕ, ℤ, ℝ ) is also often used for these special number sets. Infinite sets come in different sizes!Infinite sets come in different sizes!

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter9 Venn/Euler Diagrams John Venn

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter10 Warning: such diagrams come in different flavours (e.g., Venn or Euler). We will ‘mix and match’ flavours – This is ok as long as it’s clear what we mean.Warning: such diagrams come in different flavours (e.g., Venn or Euler). We will ‘mix and match’ flavours – This is ok as long as it’s clear what we mean.

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter11 Basic Set Relations: Member of x  S ( “ x is in S ” ) is the proposition that object x is an  lement or member of set S.x  S ( “ x is in S ” ) is the proposition that object x is an  lement or member of set S. –e.g. 3  N, “ a ”  {x | x is a letter of the alphabet} Set equality is defined in terms of  : S=T :  def  x: x  S  x  T “ Two sets are equal iff they have the same members. ”Set equality is defined in terms of  : S=T :  def  x: x  S  x  T “ Two sets are equal iff they have the same members. ” Notation: x  S :  def  (x  S)Notation: x  S :  def  (x  S)

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter12 A set can be empty Suppose we call a set S empty iff it has no elements:  x(x  S).Suppose we call a set S empty iff it has no elements:  x(x  S). Prove that  xy((empty(x)  empty(y)  x=y)Prove that  xy((empty(x)  empty(y)  x=y) Note: this formula quantifies over sets!Note: this formula quantifies over sets!

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter13 There ’ s only one empty set Prove that  xy((empty(x)  empty(y))  x=y) Proof by Reductio ad Absurdum: Suppose there existed a and b such that empty(a) and empty(b).Suppose there existed a and b such that empty(a) and empty(b). Thus,  x(x  a)   x(x  b)Thus,  x(x  a)   x(x  b) Suppose a  b. This would mean that either  x(x  a   x  b) or  x(x  b   x  a)Suppose a  b. This would mean that either  x(x  a   x  b) or  x(x  b   x  a) But the first case cannot hold, for  x(x  a). The second case cannot hold, for  x(x  b)But the first case cannot hold, for  x(x  a). The second case cannot hold, for  x(x  b) Contradiction, so QEDContradiction, so QED

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter14 The Empty Set We have seen that there exists exactly one empty set, so we can give it a name:We have seen that there exists exactly one empty set, so we can give it a name:  ( “ the empty set ” ) is the unique set that contains no elements whatsoever.  ( “ the empty set ” ) is the unique set that contains no elements whatsoever.  = {} = {x|x  x} =... = {x|False}  = {} = {x|x  x} =... = {x|False} Any set containing exactly one element is called a singletonAny set containing exactly one element is called a singleton

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter15 Subset and Superset Relations S  T ( “ S is a subset of T ” ) means that every element of S is also an element of T.S  T ( “ S is a subset of T ” ) means that every element of S is also an element of T. S  T :  def  x (x  S  x  T)S  T :  def  x (x  S  x  T) What do you think about these?What do you think about these? –  S ? –S  S ?

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter16 Subset and Superset Relations S  T ( “ S is a subset of T ” ) means that every element of S is also an element of T.S  T ( “ S is a subset of T ” ) means that every element of S is also an element of T. S  T :  def  x (x  S  x  T)S  T :  def  x (x  S  x  T) What do you think about these?What do you think about these? –  S ? Yes –S  S ? Yes

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter17 Subset and Superset Relations More notation:More notation: S  T ( “ S is a superset of T ” ) :  def T  S.S  T ( “ S is a superset of T ” ) :  def T  S. Note S=T  S  T  S  T. Note S=T  S  T  S  T. :  def  (S  T), i.e.  x(x  S  x  T) :  def  (S  T), i.e.  x(x  S  x  T)

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter18 Proper (Strict) Subsets & Supersets S  T ( “ S is a proper subset of T ” ) means that S  T but.S  T ( “ S is a proper subset of T ” ) means that S  T but. Example:{1,2}  {1,2,3}  {1,2,3}, We have {1,2,3}  {1,2,3}, but not but not {1,2,3}  {1,2,3}

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter19 Sets Are Objects, Too! The elements of a set may themselves be sets.The elements of a set may themselves be sets. E.g. let S={x | x  {1,2,3}} then S = …E.g. let S={x | x  {1,2,3}} then S = …

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter20 Sets Are Objects, Too! The objects that are elements of a set may themselves be sets.The objects that are elements of a set may themselves be sets. E.g. let S={x | x  {1,2,3}} then S={ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}E.g. let S={x | x  {1,2,3}} then S={ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Note that 1  {1}  {{1}}Note that 1  {1}  {{1}}

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter21 Cardinality and Finiteness |S| (read “ the cardinality of S ” ) is a measure of how many different elements S has.|S| (read “ the cardinality of S ” ) is a measure of how many different elements S has. E.g., |  |=0, |{1,2,3}| = 3, |{a,b}| = 2, |{{1,2,3},{4,5}}| = ____E.g., |  |=0, |{1,2,3}| = 3, |{a,b}| = 2, |{{1,2,3},{4,5}}| = ____ If |S|  N, then we say S is finite. Otherwise, we say S is infinite.If |S|  N, then we say S is finite. Otherwise, we say S is infinite.

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter22 The Power Set Operation The power set P(S) of a set S is the set of all subsets of S. P(S) :≡ {x | x  S}.The power set P(S) of a set S is the set of all subsets of S. P(S) :≡ {x | x  S}. E.g. P({a,b}) = { , {a}, {b}, {a,b}}.E.g. P({a,b}) = { , {a}, {b}, {a,b}}. Sometimes P(S) is written 2 S, because |P(S)| = 2 |S|.Sometimes P(S) is written 2 S, because |P(S)| = 2 |S|. It turns out  S:|P(S)|>|S|, e.g. |P(N)| > |N|. There are different sizes of infinite sets!It turns out  S:|P(S)|>|S|, e.g. |P(N)| > |N|. There are different sizes of infinite sets!

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter23 Review: Set Notations So Far Set enumeration {a, b, c}Set enumeration {a, b, c} and set-builder {x|P(x)}. and set-builder {x|P(x)}.  relation, and the empty set .  relation, and the empty set . Set relations =, , , , , , etc.Set relations =, , , , , , etc. Venn diagrams.Venn diagrams. Cardinality |S| and infinite sets N, Z, R.Cardinality |S| and infinite sets N, Z, R. Power sets P(S).Power sets P(S).

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter24 Ordered n-tuples These are like sets, except that duplicates matter, and the order makes a difference.These are like sets, except that duplicates matter, and the order makes a difference. For n  N, an ordered n-tuple or a sequence of length n is written (a 1, a 2, …, a n ). Its first element is a 1, etc.For n  N, an ordered n-tuple or a sequence of length n is written (a 1, a 2, …, a n ). Its first element is a 1, etc. Note that (1, 2)  (2, 1)  (2, 1, 1).Note that (1, 2)  (2, 1)  (2, 1, 1). Empty sequence, singlets, pairs, triples, …, n-tuples.Empty sequence, singlets, pairs, triples, …, n-tuples. Contrast with sets: {...}

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter25 n-tuples have many applications. For example,n-tuples have many applications. For example,

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter26 Relations are often spelled out by means of n- tuples. E.g., here are two 2-place relations: < = { (0,1), (1,2), (0,2), …) } Like-to-watch = {(John,news),(Mary,soap),(Ellen,movies)} The first and second argument of a relation may come from different sets, e.g. first: element of the set of persons second: element of the set of TV-programs

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter27 Cartesian Products of Sets For sets A, B, their Cartesian product A  B :  {(a, b) | a  A  b  B }.For sets A, B, their Cartesian product A  B :  {(a, b) | a  A  b  B }. E.g. {a,b}  {1,2} = {(a,1),(a,2),(b,1),(b,2)}E.g. {a,b}  {1,2} = {(a,1),(a,2),(b,1),(b,2)} {John,Mary,Ellen}x{News,Soap}={John,Mary,Ellen}x{News,Soap}= René Descartes ( )

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter28 Cartesian Products of Sets For sets A, B, their Cartesian product A  B :  {(a, b) | a  A  b  B }.For sets A, B, their Cartesian product A  B :  {(a, b) | a  A  b  B }. E.g. {a,b}  {1,2} = {(a,1),(a,2),(b,1),(b,2)}E.g. {a,b}  {1,2} = {(a,1),(a,2),(b,1),(b,2)} {John,Mary,Ellen}x{News,Soap}= {(John,News),(Mary,News),(Ellen,News), (John,Soap),(Mary,Soap),(Ellen,Soap)}{John,Mary,Ellen}x{News,Soap}= {(John,News),(Mary,News),(Ellen,News), (John,Soap),(Mary,Soap),(Ellen,Soap)} If R is a relation between A and B then R  AxBIf R is a relation between A and B then R  AxB

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter29 Cartesian Products of Sets Note thatNote that –for finite A, B, |A  B| = |A|.|B| –the Cartesian product is not commutative: i.e.,  AB: A  B=B  A. –notation extends naturally to A 1  A 2  …  A n

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter30 Review of §1.6 Sets S, T, U… Special sets N, Z, R.Sets S, T, U… Special sets N, Z, R. Set notations {a,b,...}, {x|P(x)}…Set notations {a,b,...}, {x|P(x)}… Set relation operators x  S, S  T, S  T, S=T, S  T, S  T. (These form propositions.)Set relation operators x  S, S  T, S  T, S=T, S  T, S  T. (These form propositions.) Finite vs. infinite sets.Finite vs. infinite sets. Set operations |S|, P(S), S  T.Set operations |S|, P(S), S  T. Next up: §1.5: More set ops: , , .Next up: §1.5: More set ops: , , .

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter31 Start §1.7: The Union Operator For sets A, B, their  nion A  B is the set containing all elements that are either in A, or ( “  ” ) in B (or, of course, in both).For sets A, B, their  nion A  B is the set containing all elements that are either in A, or ( “  ” ) in B (or, of course, in both). Formally,  A,B: A  B = {x | x  A  x  B}.Formally,  A,B: A  B = {x | x  A  x  B}. Note that A  B is a superset of both A and B (in fact, it is the smallest such superset):  A, B: (A  B  A)  (A  B  B)Note that A  B is a superset of both A and B (in fact, it is the smallest such superset):  A, B: (A  B  A)  (A  B  B)

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter32 {a,b,c}  {2,3} = {a,b,c,2,3}{a,b,c}  {2,3} = {a,b,c,2,3} {2,3,5}  {3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}{2,3,5}  {3,5,7} = {2,3,5,3,5,7} ={2,3,5,7} Union Examples

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter33 The Intersection Operator For sets A, B, their intersection A  B is the set containing all elements that are simultaneously in A and ( “  ” ) in B.For sets A, B, their intersection A  B is the set containing all elements that are simultaneously in A and ( “  ” ) in B. Formally,  A,B: A  B={x | x  A  x  B}.Formally,  A,B: A  B={x | x  A  x  B}. Note that A  B is a subset of both A and B (in fact it is the largest such subset):  A, B: (A  B  A)  (A  B  B)Note that A  B is a subset of both A and B (in fact it is the largest such subset):  A, B: (A  B  A)  (A  B  B)

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter34 {a,b,c}  {2,3} = ___{a,b,c}  {2,3} = ___ {2,4,6}  {3,4,5} = ______{2,4,6}  {3,4,5} = ______ Intersection Examples Think “ The intersection of University Ave. and W 13th St. is just that part of the road surface that lies on both streets. ”  {4}

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter35 Disjointness Two sets A, B are called disjoint (i.e., not joined) iff their intersection is empty. (A  B=  )Two sets A, B are called disjoint (i.e., not joined) iff their intersection is empty. (A  B=  ) Example: the set of even integers is disjoint with the set of odd integers.Example: the set of even integers is disjoint with the set of odd integers.

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter36 Inclusion-Exclusion Principle How many elements are in A  B? Can you think of a general formula? (Express in terms of |A| and |B| and whatever else you need.)How many elements are in A  B? Can you think of a general formula? (Express in terms of |A| and |B| and whatever else you need.)

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter37 Inclusion-Exclusion Principle How many elements are in A  B? |A  B| = |A|  |B|  |A  B|How many elements are in A  B? |A  B| = |A|  |B|  |A  B| Example: How many students are on our class list? Consider set E  I  M, I = {s | s turned in an information sheet} M = {s | s sent the TAs their address}Example: How many students are on our class list? Consider set E  I  M, I = {s | s turned in an information sheet} M = {s | s sent the TAs their address} Some students may have done both! |E| = |I  M| = |I|  |M|  |I  M|Some students may have done both! |E| = |I  M| = |I|  |M|  |I  M|

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter38 Set Difference For sets A, B, the difference of A and B, written A  B, is the set of all elements that are in A but not B. Formally: A  B :   x  x  A  x  B For sets A, B, the difference of A and B, written A  B, is the set of all elements that are in A but not B. Formally: A  B :   x  x  A  x  B  Also called: The complement of B with respect to A.Also called: The complement of B with respect to A.

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter39 Set Difference Examples {1,2,3,4,5,6}  {2,3,5,7,9,11} = ___________{1,2,3,4,5,6}  {2,3,5,7,9,11} = ___________ Z  N  {…, −1, 0, 1, 2, … }  {0, 1, … } = {x | x is an integer but not a nat. #} = {x | x is a negative integer} = {…, −3, −2, −1}Z  N  {…, −1, 0, 1, 2, … }  {0, 1, … } = {x | x is an integer but not a nat. #} = {x | x is a negative integer} = {…, −3, −2, −1} {1,4,6}

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter40 Set Difference - Venn Diagram A−B is what ’ s left after B “ takes a bite out of A ”A−B is what ’ s left after B “ takes a bite out of A ” Set A Set B Set A  B Chomp!

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter41 Set Complements The universe of discourse can itself be considered a set, call it U.The universe of discourse can itself be considered a set, call it U. When the context clearly defines U, we say that for any set A  U, the complement of A, written, is the complement of A w.r.t. U, i.e., it is U  A.When the context clearly defines U, we say that for any set A  U, the complement of A, written, is the complement of A w.r.t. U, i.e., it is U  A. E.g., If U=N,E.g., If U=N,

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter42 Set Identities A A 

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter43 Set Identities A  = AA  = A A  U =A  U =

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter44 Set Identities A  = AA  = A A  U = AA  U = A

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter45 Set Identities A  = A = A  UA  = A = A  U A  U = U A  = A  U = U A  = 

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter46 Set Identities A  = A = A  UA  = A = A  U A  U = U A  = A  U = U A  =  A  A = A = A  AA  A = A = A  A A  B = B  A A  B = B  AA  B = B  A A  B = B  A A  (B  C)=(A  B)  C A  (B  C)=(A  B)  CA  (B  C)=(A  B)  C A  (B  C)=(A  B)  C

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter47 Have you seen similar patterns before?

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter48 Read:  := ,  := ,  :=F, U:=T A  = A = A  UA  = A = A  U A  U = U, A  = A  U = U, A  =  A  A = A = A  AA  A = A = A  A A  B = B  A, A  B = B  AA  B = B  A, A  B = B  A A  (B  C)=(A  B)  C, A  (B  C)=(A  B)  CA  (B  C)=(A  B)  C, A  (B  C)=(A  B)  C

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter49 Set Identities (don ’ t worry about their names) Identity: A  = A = A  UIdentity: A  = A = A  U Domination: A  U = U, A  = Domination: A  U = U, A  =  Idempotent: A  A = A = A  AIdempotent: A  A = A = A  A Double complement:Double complement: Commutative: A  B = B  A, A  B = B  ACommutative: A  B = B  A, A  B = B  A Associative: A  (B  C)=(A  B)  C, A  (B  C)=(A  B)  CAssociative: A  (B  C)=(A  B)  C, A  (B  C)=(A  B)  C

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter50 DeMorgan ’ s Law for Sets Exactly analogous to (and provable from) DeMorgan ’ s Law for propositions.Exactly analogous to (and provable from) DeMorgan ’ s Law for propositions.

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter51 Proving Set Identities To prove statements about sets, of the form E 1 = E 2 (where the Es are set expressions), here are three useful techniques: 1. Use equivalence laws 2. Prove E 1  E 2 and E 2  E 1 separately. 3. Use a membership table.

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter52 Method 2: Mutual subsets Example: Show A  (B  C)=(A  B)  (A  C).

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter53 Method 2: Mutual subsets Example: Show A  (B  C)=(A  B)  (A  C). Part 1: Show A  (B  C)  (A  B)  (A  C).Part 1: Show A  (B  C)  (A  B)  (A  C). –Assume x  A  (B  C), & show x  (A  B)  (A  C). –We know that x  A, and either x  B or x  C. Case 1: x  B. Then x  A  B, so x  (A  B)  (A  C).Case 1: x  B. Then x  A  B, so x  (A  B)  (A  C). Case 2: x  C. Then x  A  C, so x  (A  B)  (A  C).Case 2: x  C. Then x  A  C, so x  (A  B)  (A  C). –Therefore, x  (A  B)  (A  C). –Therefore, A  (B  C)  (A  B)  (A  C). Part 2: Show (A  B)  (A  C)  A  (B  C). (analogous)Part 2: Show (A  B)  (A  C)  A  (B  C). (analogous)

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter54 Mutual subsets A variant of this method: translate into propositional logic, then reason within propositional logic, then translate back into set theory. E.g.,A variant of this method: translate into propositional logic, then reason within propositional logic, then translate back into set theory. E.g., Show A  (B  C)  (A  B)  (A  C). Suppose x  A  (x  B  x  C). Prove (x  A  x  B)  (x  A  x  C).Show A  (B  C)  (A  B)  (A  C). Suppose x  A  (x  B  x  C). Prove (x  A  x  B)  (x  A  x  C).

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter55 Review of § Sets S, T, U… Special sets N, Z, R.Sets S, T, U… Special sets N, Z, R. Set notations {a,b,...}, {x|P(x)}…Set notations {a,b,...}, {x|P(x)}… Relations x  S, S  T, S  T, S=T, S  T, S  T.Relations x  S, S  T, S  T, S=T, S  T, S  T. Operations |S|, P(S), , , , ,Operations |S|, P(S), , , , , Set equality proof techniquesSet equality proof techniques

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter56 Generalized Unions & Intersections Since union & intersection are commutative and associative, we can extend them from operating on ordered pairs of sets (A,B) to operating on sequences of sets (A 1,…,A n ), or even on unordered sets of sets, X={A | P(A)} (for some property P).Since union & intersection are commutative and associative, we can extend them from operating on ordered pairs of sets (A,B) to operating on sequences of sets (A 1,…,A n ), or even on unordered sets of sets, X={A | P(A)} (for some property P). (This is just like using  when adding up large or variable numbers of numbers) (This is just like using  when adding up large or variable numbers of numbers)

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter57 Generalized Union Binary union operator: A  BBinary union operator: A  B n-ary union: A 1  A 2  …  A n :  ((…((A 1  A 2 )  …)  A n ) (grouping & order is irrelevant)n-ary union: A 1  A 2  …  A n :  ((…((A 1  A 2 )  …)  A n ) (grouping & order is irrelevant) “ Big U ” notation:“ Big U ” notation: Or for infinite sets of sets:Or for infinite sets of sets:

Module #3 - Sets 10/27/2015Michael P. Frank / Kees van Deemter58 Generalized Intersection Binary intersection operator: A  BBinary intersection operator: A  B n-ary intersection: A 1  A 2  …  A n  ((…((A 1  A 2 )  …)  A n ) (grouping & order is irrelevant)n-ary intersection: A 1  A 2  …  A n  ((…((A 1  A 2 )  …)  A n ) (grouping & order is irrelevant) “ Big Arch ” notation:“ Big Arch ” notation: Or for infinite sets of sets:Or for infinite sets of sets: