Computing Fundamentals 1 Lecture 6 A Theory of Sets

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Computing Fundamentals 1 Lecture 6A Theory of
Sets

• Lecturer Patrick Browne
• http//www.comp.dit.ie/pbrowne/
• Room K308
• Based on Chapter 11.
• A Logical approach to Discrete Math
• By David Gries and Fred B. Schneider

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Sets

• Set theory can be defined as an extension of
  predicate calculus. A set is a collection of
  distinct elements (e.g. set of integers, set of
  students).
• The operations on sets (e.g. union) correspond in
  a natural way to the logical connectives of logic
  (e.g. or). Later we will show how DeMorgans Laws
  for logic can be applied to sets.

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Sets

• Two methods of describing sets set enumeration
  and set comprehension.
• Set comprehension is defined in terms testing
  membership in sets.
• This membership test is the basis for the
  definition of equality of sets.

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Sets

• Enumerations list elements
• Constants 1,2,3,
• Variables a,b
• Comprehension state properties
• x? 0 ? x lt 5 2 ? x
• Represents
• 0,2,4,6,8
• The general form of comprehension (11.1) is
• xt R E

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Sets

• Recall from Lecture 1 a state is a list of
  variables with associated values.
• The evaluation of
• xt RE
• in a state yields the set of values from
  evaluating Ex v in the state for each value
  v in t such that Rxv holds in that state.
  Notation similar to quantification

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Universe

• A theory of sets concerns sets constructed from
  some collection of elements. There is a theory of
  sets of integers, a theory of sets of characters,
  a theory of sets of sets of integers. The
  collection of elements is called the domain of
  discourse or universe of values and is denoted by
  U. The DOD can be thought of as the type of every
  set variable in the theory (e.g. DOD is set(?),
  then vset(?)).

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Universe

• When several set theories are being used at the
  same time, there is a different universe for
  each. The name U is then overloaded similar to 1
  being considered as a real or an integer.

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Set Membership Equality

• For expression e and a set valued expression S
• e ? S
• is an expression whose value is the value of
  the statement e is a member of S. We can write
  e is not a member of S as
• ?(e ? S) or (e?S)
• The symbol ? is treated as conjunctional operator
  and has the same precedence as equality ().

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Set Membership Equality

• (11.3) Axiom, set membership. For expression Ft
  and set xR Et for type t
• provided ? occurs(x,F)
• F ? xR E ? (?x RFE)
• Example
• y ? x xgt1 3x) ? (?x xgt1y3x)
• (11.4) Axiom ,extensionality
• ST ? (?x x?S ? x?T)
• An extensional definition depends only on the
  content of the set. In contrast, an intentional1
  definition describes how the set was defined or
  constructed.

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Sets versus Predicates

• (11.7) Theorem
• x ? xRx ? R
• y ? xR ? Rxy
• Theorem (11.7) formalizes the connection between
  sets and predicates a predicate is a
  representation for the set of argument-values for
  which the predicate is true.
• 2 ? xxgt1 ? 2gt1

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Sets versus Predicates

• (11.8) The principle of comprehension
• To each predicate R there corresponds a set
  comprehension xR , which contains the objects
  t that satisfy R R is called the characteristic
  predicate of the set.

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Sets versus Predicates

• (11.8) means we can define a set using
  comprehension and by its characteristic
  predicate.
• For example the set comprehension
• S x x3 ? x5
• And the characteristic predicate
• x ? S ? x3 ? x5 (for all x)
• Are equivalent.

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Sets versus Predicates

• Theorem (11.10)
• xQxR is valid iff Q ? R is valid.

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Operations on Sets
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Operations on Sets

• Three methods for proving set equality
• (a) Use Leibniz directly
• (b) Use axiom of Extensionality (11.4) and prove
  v ? S ? v ? T for an arbitrary value of v.
• (c) Prove Q ? R and conclude xQxR

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Operations on Sets

• Let S,T,U,V have type set(t) for some type t.
  This convention allows us to shorten a definition
  
• For example, set cardinality
• (?Sset(t) S (?x x?S1))
• can be defined as
• S (?x x?S1)

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Operations on Sets
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Operations on Sets
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Operations on Sets
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Operations on Sets
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EU set

• Let the set EU be the set of all countries in the
  European Union (old).
• EU the set of all countries in the European
  Union.
• It could be declared as an enumeration
• EUA,B,DK,F,SF,D,GB,GR,IRL,I,L,NL,P,E,S
• The variable homeland can refer to one element in
  the set EU
• homeland ? EU
• beneluxB,NL,L

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The Empty Set and Singleton

• It is possible to have a set with no values, it
  is called the empty set and denoted as
• 0 or
• A set that contains only one element is called a
  singleton set. For example IRL
• Note the difference between the set IRL and
  element IRL.
• Sets are not ordered. There are no duplicates,
  each element is unique.

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Subset or Set inclusion
The following statements are true B,NL z
B,NL,L 0 zB,NL,L B,NL,L z B,NL,L
X (universal set) SPX (? is power set) TPX S z
T
T
S
X
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Subset Superset

• S is a subset of T if every element if every
  element of S is an element of T.
• (11.13) axiom, subset
• S ? T ? (?x x ? S x ? T)
• (11.14) axiom, proper subset
• S ? T ? S?T ? S?T

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Subset Superset

• (11.15) axiom, superset
• T ? S ? S?T
• (11.16) axiom, proper superset
• T ? S ? S ? T
• The axioms and theorems for set operations are
  given in the course text and the course web site,
  only a few are covered in the slides.

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Set Operators

• Set Equivalence
• Two values of the same type can be tested to see
  if they are the same by using the equals sign, as
  in
• x y
• Two sets are equal if they contain exactly the
  same elements. For example the two sets below are
  equal
• B,NL,L NL, B, L

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Set Operators

• Set Non-Equivalence
• Two values of the same type can be tested to see
  if they are not the same by using the not-equals
  sign ?. Two sets are not equal if they do not
  contain exactly the same elements
• B,NL ? B,NL,L

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Set Operators

• The membership operator is written e
• NL e B, NL, Lit is true the Netherlands is a
  Benelux country
• General case below

SPX xX xeS
.x
S
X
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Set Operators

• The non-membership operator is written
• IRL?B, NL, L Ireland is not a Benelux country

SPX xX x ? S
General case
S
.x
X
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Set Operators

• The validity of membership test.
• The value to be tested for membership must be an
  element of the underlying type of the set. For
  example
• USA e B, NL, L
• is illegal, since USA is not an element of the
  type EU.

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Set Operators

• Size Cardinality The number of values in a set
  is called its size, or cardinality, and is
  signified with the has sign
• B,NL,L 3
• IRL 1
• IRL illegal, IRL is not a set
• ? 0

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Set Operators

• Powersets The powerset of a set is written
• PS
• It is the set of all subsets of S. For example
  the powerset of the Benelux countries is
• PB,NL,L ?,
• B,NL,L,
• B,NL,B,L,NL,L,
• B,NL,L

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Set OperatorsPowersets

• When a variable is to be declared to have a type
  that is a set of elements, the type is the
  powerset of the type of the elements
• benelux PEU
• This can be read as the variable benelux is a
  subset of the set of countries EU or the
  variable benelux is a set of EU countries

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Set OperatorsPowersets

• The size of the powerset of a set is equal to two
  raised to the power of the size of the set.
• (PS) 2s
• B,NL,L 3
• (PB,NL,L) 8

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Complement

• The complement of S (S) is the set of elements
  that are not in S but are in the universe.
• (11.17) axiom, complement
• v?Sc ? v?U ? v?S

S
U
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Set Operators Union

• The union of two sets is the set containing all
  the elements that are in either the first set or
  the second set or both.

X SPX TPX T U S T U S T U S S U 0 S
S
T
X
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Set Operators Intersection

• The intersection of two sets is the set
  containing all the elements that are in the first
  set and in the second set.

X SPX TPX T I S T I S S I T S I 0 0
S
T
X
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Set Operators Difference

• The difference of two sets is the set containing
  all the elements that are in the first set and
  are not in the second set. The shaded area is the
  difference of S and T.

X SPX TPX S \ T S \ T T \ S S \ 0 S 0
\ S 0
S
T
X
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Set Operators Distributed Union

• The distributed union of a set of sets is the set
  containing just those elements that occur in at
  least one of the component sets.

X SPX RPX TPX R,S,T
R
S
T
X
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Set Operators Distributed Intersection

• The distributed intersection of a set of sets is
  the set containing just those elements that occur
  in all of the component sets.

X SPX RPX TPX R,S,T
R
T
S
X
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Disjoint Sets

• Sets that are disjoint have no elements in
  common their intersection is the empty set.
• For disjoint sets T and S the following
  expression is true
• T I S 0.

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More than 2 Disjoint Sets

• For more that two sets it becomes longer, since
  every pair must have an empty intersection. For
  example for sets A, B and C to be disjoint the
  following must be true
• A I B 0 and
• B I C 0 and
• C I A 0

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Partition

• A sequence of sets is said to partition another
  larger set if the sets are disjoint and their
  distributed union is the entire larger set if the
  sets are disjoint. For example
• if
• disjointltA,B,Cgt ? (?A,B,C T)
• then ltA,B,Cgt partition T
• For example
• ltTemporary, Part-time, Permanentgt may partition
  Employee.

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Relating sets and Boolean Expressions

• The definition of the set operations reveals a
  connection between the set operators and the
  propositional operators. For example, in the
  definition of ? , as the phrase x? of the LHS
  is distributed inward to the operands S and T of
  the RHS, ? becomes ?.
• x ? S ? T ? x ? S ? x ? T

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Relating sets and Boolean Expressions

• The properties of propositional operators are
  reflected as similar properties of set operators,
  e.g. absorption laws
• S ? (S ? T) ? S (type S,TBoolean)
• S ? (S ? T) S (type S,Tset(t))
• The zero laws for props. sets.
• S ? false ? false (type SBoolean)
• S ? 0 0 (type Sset(t))

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Set expression Es Propositional expression Ep
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Numerical Problem for 3 sets (see lab 4)

• Out of group of 42 students
• 3 studied French, Business and Music.
• 10 studied French and Business.
• 6 studied French and Music.
• 5 studied Business and Music.
• 18 studied French.
• 20 studied Business.
• 17 studied Music.

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Another Set Problem

• A group of 41 students study Finance, Business
  and Mathematics as follows
• 2 studied Finance, Business and Mathematics.
• 9 studied Finance and Business.
• 4 studied Finance and Mathematics.
• 5 studied Business and Mathematics.
• 17 studied Finance.
• 20 studied Business.
• 16 studied Mathematics.

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Another Set Problem
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Another Numerical Set Problem

• How many studied Finance and Mathematics but not
  Business?
• (Finance Mathematics) All Three
• 4 - 2 2.

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Another Set Problem

• How many studied Business and neither Finance nor
  Mathematics?
• Business - FinanceBusiness -
  MathematicsBusiness All three
• 20 5 9 2 8

The overlap (intersection) between these two
pairs was removed twice so it needs to be added
back
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Another Set Problem

• How many are taking none of the subjects?
• 17201653 subjects with intersections
• (945 ) pair wise intersections
• 53 - (9 4 5 ) 2 37 remove intersections
  and double counting
• 41 37 4 not taking a subject.

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A Set Proof

• We will prove the distribution of set union over
  set intersection
• A ? (B ? C) (A?B) ? (A?C)
• We will use the following theorem and axiom
• (1) p ?(q?r) ? (p?q)?(p?r)
• (2) S T ? (?v v ? S ? v ? T)
• And the definitions of union and intersection
• (3) v ? B ? C ? v ? B ? v ? C
• (4) v ? B ? C ? v ? B ? v ? C

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A Set Proof

• Axiom are numbered as in the previous slide. We
  prove distribution by using Extensionality (3) to
  show that an arbitrary element v is in LHS
  exactly when it is in the RHS.
• v ? A ? (B ? C)
• lt definition of ? (3) gt
• v ? A ? v ? (B ? C)
• lt definition of ? (4) gt
• v ? A ? (v ? B ? v ? C)
• lt Distribution of ? over ? (1) gt
• (v ? A ? v ? B) ? (v ? A ? v ? C)
• lt definition of ? (3) twicegt
• (v ? A ? B) ? (v ? A ? C)
• lt definition of ? (4) gt
• v ? (A ? B) ? (A ? C)

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DeMorgans First Law for sets not(A ? B)
not(A) ? not(B)
A ? B
not(A ? B)
A
A
B
B
not(A)
not(B)
A
A
B
B
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DeMorgans First Law for sets not(A ? B)
not(A) ? not(B)
not(B)
not(A)
A
A
B
B
not(A) ? not(B)
A
B
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Example Proof

• Prove (S - T) ? S ? T
• x ? (S T)
• lt Set difference 11.22 gt
• x ? S ? x ? T
• lt using x ? (S) ? x ? S gt
• x ? S ? x ? T
• lt Set Intersection 11.21gt
• x ? (S ? T)

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Set Comprehension Example

• The set of positive integers that are less than 5
  
• Solution
• x? 0 ? x ? 4x or x? 0 ? x lt5x
• Or
• 0,1,2,3,4

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Set Comprehension Example

• The set of positive integers which are divisible
  by 3 and less than 7.
• Solution
• x? 0ltxlt7 ? x/3x?3
• where / is ordinary division, ? is integer
  division
• All powers of 2.
• Solution i? 0?i2i