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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

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.

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.

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

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

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(?)).

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.

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 ().

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.

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

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.

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.

Sets versus Predicates

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

Operations on Sets

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

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)

Operations on Sets

Operations on Sets

Operations on Sets

Operations on Sets

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

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.

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

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

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.

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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.

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

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.

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

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))

Set expression Es Propositional expression Ep

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.

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.

Another Set Problem

Another Numerical Set Problem

- How many studied Finance and Mathematics but not

Business? - (Finance Mathematics) All Three
- 4 - 2 2.

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

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.

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

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)

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

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

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)

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

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