sets goal: introduce the basic terminology of set theory

Sets

Goal: Introduce the basic terminology of set theory

Copyright © Peter Cappello 2

Definition

• Visualize a dictionary as a directed graph.

• Nodes represent words

• If word w is defined in terms of word u, draw an edge from w to u.

• Can the dictionary be infinite?

• Can the dictionary have cycles?

• Thus, some words are not formally defined.

• “Set” is a primitive concept in mathematics: It is not formally defined.

• A set intuitively is an unordered collection of elements.


Preliminaries

The universe of discourse, denoted U,

intuitively is a set describing the context for

the duration of a discussion.

E. g., U is the set of integers.

(As far as I can tell, its purpose is to ensure

that the complement of a set is a set.)


Preliminaries

A set S is well-defined when we can decide

whether any particular object in the

universe of discourse is an element of S.

– S is the set of all even numbers

– S is the set of all human beings

Do we have a rule that lets us decide whether some

blob of protoplasm is a human being?


Definitions & Conventions• A set’s objects are called its members or elements.

• The symbol “” means “is a member of”.

• We can describe a set with set builder notation.

Let O denote the set x such that x is an odd positive integer

less than 10.

O = { x | x is an odd positive integer < 10 }.

Let O denote the set x of positive integers such that x < 10

and x is odd.

O = { x Z+ | x < 10 and x is odd }.

By Convention


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.

Q = { p/q | p, q Z and q 0 } is the set of Rationals.

R = the set of real numbers.


Definitions

• Set A is a subset of set B, denoted A B, when x

( x A x B ).

• Set A equals set B when they have the same

elements:

A = B when x ( x A x B ).

• We can show A = B via 2 implications:

A B B A

x ( ( x A x B ) ( x B x A ) ).


• The empty set, denoted , is the set with no elements.

• Let A be a set.• True, false, or maybe?

1. A.

2. A.

3. A A.

4. A A.

• If A B A B then A is a proper subset of B, denoted A B.


Venn Diagrams

• Venn diagram of A B. U

A

B


Cardinality

If S is a finite set with n elements, then its

cardinality is n, denoted |S|.


The Power Set

The power set of set S, denoted P(S), is { T | T S }.

• What is P( { 0, 1 } )?• What is P( )?• What is P( P( ) )?• Define

P1( S ) = P( S ),Pn( S ) = P ( Pn-1( S ) ).

• | Pn ( ) | = ?

• P1( ) = { }

• P2( ) = P( { } ) = { , { } }

• P3( ) = P( { , { } } )

= { , { }, {{ }}, { , { } } }

• | P1 ( ) | = 20

• | Pn ( ) | = 2| Pn-1 ( ) |

• Express | P5 ( ) | using only digits 2 & 0.



Cartesian Products

• The Cartesian product of sets A & B, denoted

A x B, is

A x B = { ( a, b ) | a A b B }.

• Let S = { small, medium, large }

C = { pink, lavender }.– Enumerate the ordered pairs in S x C.

– Enumerate the ordered pairs in C x S.

– Enumerate the ordered pairs in x S.

– | S x C | = ?


Cartesian Products

• Cartesian product of n sets:

A1 x A2 x … x An = { ( a1, a2, …, an ) | a1 A1, a2 A2, …, an

An }.

• Assuming the sets are finite, describe

| A1 x A2 x … x An | in terms of the cardinalities of the

component sets.

• Using sets S & C as previously described, describe

( S x C ) x ( C x S ).

• | ( S x C ) x ( C x S ) | = ?


Using Set Notation with Quantifiers

• A shorthand for x ( x R x2 ≥ 0 ) is

x R ( x2 ≥ 0 )

• A shorthand for x ( x Z x2 = 1 ) is

x Z ( x2 = 1 )

• The statements above are either true or false.

• What if you want the set of elements that make a

proposition function true?


Truth Sets of Proposition Functions

• Let P be a proposition function,

D a domain.

• The truth set of P with respect to D is

{ x D | P( x ) }.

• Enumerate the truth set

{ x N | ( x < 20 ) ( x is prime ) }.

sets goal: introduce the basic terminology of set theory

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