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Set Theory. Section 2.1 Tuesday 20 May. Definition: A set is an unordered collection of objects, called the elements or members of the set. A set is said to contain its elements. The definition does not require any relationship among the members of a set.

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set theory

Set Theory

Section 2.1

Tuesday 20 May

SETS

what is a set
Definition: A set is an unordered collection of objects, called the elements or members of the set. A set is said to contain its elements.

The definition does not require any relationship among the members of a set.

In a set, repeated elements are ignored.

Notation:

x Sdenotes “x is an element of the set S”

x S denotes “xis not an element of the set S”

What is a Set?

SETS

describing a set
Enumerate it members, enclosed by curly braces (‘{‘ and ‘}’). Examples:

vowels = { ‘a’, ‘e’, ‘i’, ‘o’, ‘u’, ‘y’}

weekdays = {“Mon”, “Tue”, “Wed”, “Thu”, “Fri”}

N = { 0, 1, 2, … }

Setbuilder notation: describe how to “build” the set

{ exp | predicate } or { expU | predicate }

evens = { y | y is even and yN }

evens = { yN | y is even }

evens = { y | n ( nN y = 2n )}

evens = { 2y | yN }

Describing a Set

SETS

some important sets
The set of Natural Numbers:N= {0, 1, 2, …}

The set of Integers:Z= { …, -2, -1, 0, 1, 2, …}

The set of Positive Integers: Z+ = {1, 2, …}

The set of Rational Numbers:Q= { p/q | p  Zq  Zq ≠ 0 }

The set of Real Numbers: R

Some Important Sets

SETS

set equality
Definition: Two sets are equal if and only if they contain the same elements.

Formally, let S and X be sets. Then S = X if and only if the following proposition is true:[x (x S→ x X )][x (x X → x S)]

Set Equality

SETS

other special sets
The universal set, denoted U, is the set of all possible elements under consideration.

The emptyset, denoted { } or Ø, is the set containing no elements.

Note that {Ø} is not the same as Ø.

Why not?????

Other Special Sets

SETS

venn diagram
An informal way to picture a set is using Venn diagrams.

Here, we are displaying the set {1,3,5}

U

  • 3
  • 5
Venn Diagram

SETS

subsets of a set
Definition: A set A is a subset of a set B, written A B, if and only if every element of A is also an element of B.

Exercise: How would you express this in Pred. Logic?

A B  x (xA → x  B)

Ais aproper subsetof B, writtenA B, if and only if (A B) (A  B ).

Exercise: Express A B as a predicate logic formula.

A B  ( x (xA → x  B)  ∃x (x∉A x  B) )

One way to show that A = B is to show that A  B and B  A.

That is: A = B (A B) (B  A)

Subsets of a Set

SETS

subsets
The empty set Ø is a subset of all sets:

S (Ø  S )

How would you prove this proposition?

Every set is a subset of itself:

S ( S  S )

How would you prove this proposition?

Subsets ….

SETS

cardinality of a set
Definition: Let S be a set. If there are exactly n (distinct) elements in S, we say that S is a finite set, and that n is the cardinality of S, written |S|.

vowels = { ‘a’, ‘e’, ‘i’, ‘o’, ‘u’, ‘y’ } is finite and |vowels| = 6.

weekdays = {“Mon”, “Tue”, “Wed”, “Thu”, “Fri”} is finite and |weekdays| = 5.

Definition:A set is infinite if it is not finite.

The set of natural numbers N is infinite

The set of real numbers R is infinite

Cardinality of a Set

SETS

power set of a set
Definition:Given a set S, the power set of S, denotedP(S), is the set of all subsets of S.

Example:P({1,2,5}) = {Ø, {1},{2},{5}, {1,2},{1,5},{2,5}, {1,2,5}}

Exercise:

If |S| = 0, what is|P(S)|?

If |S| = 1, what is|P(S)|?

If |S| = 2, what is|P(S)|?

If |S| = 3, what is|P(S)|?

What do you conjecture more generally about |P(S)| ?

Power Set of a Set

SETS

tuples
Definition:An ordered n-tuple, written (a1, a2, …, an), is the ordered collection that has a1 as its first element, a2as its second element,…, and an as its nth element.

Two n-tuples are equal if and only if each corresponding pair of their elements is equal.

(a1, a2, …, an) = (b1, b2, …, bn)  a1 = b1  a2 = b2  …  an= bn

Tuples

SETS

tuples1
Tuples

2-tuples are called (ordered) pairs.

(a, b) = (c, d)iff a = c and b = d

(a, b)≠(b, a)unless a = b

3-tuples are called (ordered) triples.

(a, b, c) = (d, e, f )iff a = d and b = e and c = f

(a, b, c)≠(c, b, a)unless a = c

13

SETS

cartesian product
Definition:Let A and B be sets. The Cartesian product of A andB, denoted A × B, is the set of all ordered pairs (a, b) where aA andbB.

That is:A × B ={ (a, b)| aAbB}

Given:weekdays = {“Mon”, “Tue”, “Wed”, “Thu”, “Fri” }amStartTimes = { “8:00a”, “9:00a”, “10:00a”, “11:00a” }

A calendar for scheduling morning meetings:

weekdays × amStartTimes

IfA ≠ and B ≠ , then(A × B = B × A)  (A=B)

How would you prove this?

Cartesian Product

SETS

cartesian product1
Cartesian Product

Definition:Let A1, A2, …, An be sets. The Cartesian product of A1, A2, …, An, denoted A1× A2 × … × An, is the set of all ordered n-tuples (a1, a2, …, an ) where aiAi fori = 1, 2, …, n.

That is:A1× A2 × …× An={ (a1, a2, …, an )| aiAi fori = 1, …, n}

Example:

Given: weekdays, amStartTimes, as defined previously, andcseLabs = { PRIP, SENS, ELANS, LINKS, METLAB, DEVOLAB }

All possible triples indicating a potential morning start time and day for a meeting of a CSE lab: weekdays ×amStartTimes×cseLabs

15

SETS

relation
Relation

Every subset of a Cartesian product is a Relation

One possible schedule for the CSE conference room:{(“Mon”, “9:00a”, PRIP), (“Mon”, “11:00a”, SENS), (“Tue”, “10:00a”, GEM), (“Thu”, “10:00a”, GEM), (“Thu”, “11:00a”, ELAN) }weekdays × amStartTimes × cseLabs

An example binary relation:{(0,0), (1,1), (2,4), (3,9), (4, 16), …}  N × N

What well-know function does this relation describe?

sq = { (n, n2) | n N }

Chapter 8 is all about relations

16

SETS

set operations

Set Operations

Section 2.2

SET THEORY

set union

A

B

U

Set Union
  • Definition: Let A and B be sets. The unionof A and B, denoted A B, is the set containing those elements that are either in A or in B, or in both.
  • Formally, A B = {x | xA  xB}
  • Venn Diagram

SET THEORY

set intersection

U

A

B

Set Intersection
  • Definition: Let A and B be sets. The intersection of A and B, denoted by A ∩B, is the set containing those elements which are in both A and B.
  • Formally, A ∩B = { x | xAxB }

SET THEORY

set complement
Set Complement
  • Definition: Let U be the universal set. The complement of a set S, denoted S,is the set containing elements of U which are not in S.
  • Formally, S= { x  U | x S }
  • Venn Diagram:

U

S

S

SET THEORY

set difference

A

Set Difference
  • Definition:Let A andB be sets. The differenceof A and B, denoted A ‒ B, is the set containing those elements that are in A but are not in B.
    • A ─Bis the complement of B with respect to A.
    • A ─B={x | xAx B}

U

A – B

B

SET THEORY

exercise
Exercise:

Let:n range over the natural numbers NA = { 6n | 6n< 41 }B = { 9n | 9n< 41 }

Find the following:

AB

A∩B

A – B

A∩B

A × B

SET THEORY

membership table for set operators
Membership table for set operators
  • Represent conditions for membership in table form

. . .

1= an element in the set 0 = an element is not in the set

What similarities do you notice between membership tables and truth tables?

SET THEORY

set identities
Set Identities
  • Setstogether withtheoperators { , ∩, ∪} define a special class of Boolean algebraic structures.

SET THEORY

proving set identities example
Proving Set Identities – Example.
  • (A B) = A B
  • To show this identity we can:
    • Use the definition thatS = T (S T ) (T  S) (x S)  (x T)
    • Use Set Builder notation
    • Use membership tables

SET THEORY

example using membership table
Example – Using Membership Table

1= an element in the set 0 = an element is not in the set

SET THEORY

example we can use proven identities to prove new identities
ExampleWe can use proven identities to prove new identities.

Thus, we have proved that:

(A (BC)) = (AB) (AC)

SET THEORY

proof of set identities
Proof of Set Identities…
  • We can use Venn diagrams to get some ideas about the correctness of identities.
  • But, we cannotprove identities using Venn diagrams.

SET THEORY

generalized union
Generalized Union
  • Definition:The union of a collection (set) of sets is the set containing those elements that are members of at least one set in the collection.
    • In the case of a finite collection of sets,A0, A1, …, An-1:
  • In the case of an infinite collection of sets, A0, A1, …:where N is the set of natural number, N = {0, 1, 2, …}.

SET THEORY

example
Example
  • Let Ai = { i, i + 1, i + 2, ….}.

SET THEORY

generalized intersection
Generalized Intersection
  • Definition:Theintersection of a collection (set) of sets is the set containing those elements that are members of all the sets in the collection.
    • In the case of a finite collection of sets,A0, A1, …, An:
  • In the case of an infinite collection of sets, A0, A1, …:

SET THEORY

example1
Example
  • Let Ai = { i, i + 1, i + 2, ….}.

SET THEORY

exercise1
Exercise
  • Let
  • Find

SET THEORY

slide37

Disjoint Sets

  • Definition: Two sets A and B are disjointif their intersection is the empty set, i.e., if A ∩B =Ø.
  • Definition: The sets A0, A1, …, An-1 are pairwise disjoint if each pair of them is disjoint, i.e., if, for all 0 ≤i, j ≤ n-1, Ai ∩Aj= Ø unless i= j .
  • Observation: If A0, A1, …, An-1 are finite sets, then the sets are pairwise disjoint and only if
    • Assume this “observation” is an axiom for now.
    • To prove it we need a formal definition for the cardinality of a set (see next section).

SET THEORY

principle of inclusion exclusion 1
Principle of inclusion-exclusion 1:

Set identities can be

be proved using any

of the three methods

just discussed

(exercise).

  • If A and B are finite sets, then

|A B| = |A| + |B| - |A ∩B|

Proof: As A  B = (A – B)  (A ∩ B) (B – A)and the sets on the right side are pairwise disjoint, the previous observation implies |A  B| = |A – B| + |A ∩ B|+ |B – A| (*).

U

A

B

Moreover, as A = (A – B)  (A ∩ B) and B = (B – A)  (A ∩ B), this

same observation implies that |A| = |A – B| + |A ∩ B| (**) and that

|B| = |B – A| + |A ∩ B| (***). Solving (**) and (***) for |A – B|

and |B – A|, respectively, and substituting them in (*), we get:|A  B| = (|A| – |A ∩ B|) + |A ∩ B|+ (|B| – |A ∩ B|) = |A| + |B| – |A ∩ B|.

SET THEORY

russell s paradox
Russell’s Paradox
  • Let S be the set of all sets which do not contain themselves.

That is, S = { T | T  T }

  • Does S contain itself???

SET THEORY

computer representation of finite sets
Computer Representation of Finite Sets
  • Use an arbitrary ordering of the elements of the finite universal set U: (a0, a1, …, an-1)
  • Represent a subset A of U with the bit string of length n: ith bit is: 1, ifai  A;0, if ai  A
  • Example:For A = { 6n | n< 40 } and B = { 9n | n< 40 }, we can use strings of length 40:A: (1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,…) B: (1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,…)
  • What string represents Ø? What string represents U?

SET THEORY

computer representation
Computer Representation…
  • With the string representation of a set, and the following conventions, we can do any operation
    • Set complementation: Invert each bit (0 to 1, 1 to 0)
    • Set union: Do a bit-wise “addition”
      • 0 + 1 = 1 + 0 = 1
      • 1 + 1 = 1, 0 + 0 = 0
    • Set Intersection: Do a bit-wise “multiplication”
      • 0 × 1 = 1 × 0 = 0
      • 1 × 1 = 1, 0 × 0 = 0

SET THEORY

exercise2
Exercise:

Using:A: (1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0) B: (1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0)

Find the following:

AB

A∩B

A – B

A∩B

SET THEORY

slide43

Note on computer implementations

  • Some programming languages have the finite set as a standard data type.
  • The Standard Template Library of C++ has a set class. (The multiset allows repeated elements.)
  • Real numbers (floats, doubles) typically cannot be members of practical sets because uniqueness ( = test) is not well defined.
  • Set ops are typically O(log n) or better in execution time, where n is the set size

SET THEORY