1. Sets

2. Copyright © Peter Cappello 2011
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.
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3. Preliminaries
The universe of discourse, denoted U, intuitively is a set describing the context for the duration of a discussion.
E. g., let U be the set of integers.
(As far as I can tell, the purpose of the Universe is to ensure that the complement of a set is a set.)
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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?
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5. Definitions & Conventions
A set’s objects are called its members or elements.
We can describe a set with set builder notation.
O = { x | x is an odd positive integer < 10 }.
O = { x  Z+ | x < 10 and x is odd }.
By convention:
N = { 0, 1, 2, 3, … } the set of natural numbers.
Z = { …, -2, -1, 0, 1, 2, … } the set of integers.
Z+ = { 1, 2, 3, … } the set of positive integers.
Q = { p/q | p, q  Z and q  0 } the set of rationals.
R = the set of real numbers.
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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 ) ).
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The empty set, denoted , is the set with no elements.
Let A be an arbitrary set.
True, false, or maybe?
  A.
  A.
A  A.
A  A.
If A  B  A  B then A is a proper subset of B, denoted A  B.
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Venn Diagrams
Venn diagram of A  B. U B A
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Cardinality
If S is a finite set with n elements, then its cardinality is n, denoted |S|.
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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(  ) )?
Let
P1( S ) = P( S ),
Pn( S ) = P ( Pn-1( S ) ), for n > 1
| Pn (  ) | = ?
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| Pn (  ) | = 2| Pn-1 (  ) |
Express | P5 (  ) | using only digits 2 & 0.
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2n is the number of vertices in a n-dimensional cube.
Are there any connections between:
an n-dimensional cube
the power set of a set with n elements?
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Cartesian Products
The Cartesian product of sets A and B, denoted A x B, is
A x B = { ( a, b ) | a  A  b  B }.
Let S = { small, medium, large } and 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 | = ?
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Cartesian Products
Generalizing Cartesian product to n sets:
A1 x A2 x … x An = { ( a1, a2, …, an ) | a1  A1, a2  A2, …, an  An }.
Describe | A1 x A2 x … x An | in terms of the cardinalities of the component sets.
Using sets S and C as previously described, describe ( S x C ) x ( C x S ).
| ( S x C ) x ( C x S ) | = ?
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17. 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?
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18. Truth Sets of Proposition Functions
Let P be a proposition function and 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 ) }.
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