A%20Quick%20Math%20Overview

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A Quick Math Overview

• (not the last one!)
• Section 1.4, you should it read carefully!
• Reading the corresponding material from the book
  is strongly encouraged

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Why Math?

• We will formulate precise definitions of
  different kinds of automata

• We will formulate and proof properties of these
  automata

• Along the way we will formulate precisely
• Nondeterminism
• Computation
• State (configuration) of a computation

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Why Math? (II)
We are going to define formal models of
computation
Name of Model
Kind of language
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Sets

• Sets are denoted by ltcollection of elementsgt
• Examples
• a,b
• 1, 2, , 100
• 0, 1, 2,
• 0,2,4,
• 2n n ? ?

the empty set
the set consisting of the elements a and b
the set consisting of the empty set
the set consisting of the first 100 natural
numbers
the set consisting of all natural numbers Also
denoted by ?.
the set of all natural pair numbers
the set of all natural pair numbers
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Set Inclusion and Set Equality
Definition Given 2 sets, A and B, A is contained
in B, denoted by A ?B, if every element in A is
also an element in B
true
true
false
Definition Given 2 sets A and B, A is equal to
B, denoted by A B, if A ? B and B ? A
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Cartesian Product
Definition Given two sets, A and B, the
Cartesian product of A and B, denoted by A ? B,
is the following set
(a,b) a ?A and b ? B

• Examples
• What is 1, 2 , 3 ? a,b
• True or false (1,a), (3,b) ? 1, 2 , 3 ?
  a,b
• True or false 1,2,3 ? 1, 2 , 3 ? a,b

(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)
true
false
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Cartesian Product II
Definition Given three sets, A, B and C, the
Cartesian product of A, B, and C denoted by A ? B
? C, is the following set
(a,b,c) a ?A, b ? B, c ? C
Definition. (x,y,z) (x,y, z) only if
x x, y y and z z
These definitions can be extended to define the
Cartesian product
A1 ? A2 ? ? An and equality between n-tuples
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Cartesian Product (2)

• More examples
• What is 1, 2 , 3 ? a,b ? ?,?
• What is the form of the set A ? B ? C ? D
• What is the form of the set A ? B ? (C ? D)
• What is the form of the set (A ? B ) ? (C ? D)

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Conventions and Precedence
( (8 ? 2) (3 7))
A ? (B ? C) ? D
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Relations
Definition Given two sets, A and B, A relation R
is any subset of A ? B. In other words, R ? A ? B

• Motivation We want to indicate which elements in
  A are related to which elements in B
• Question what does the relation A ? B indicates?

• Examples of relations in real life?

(p,s) p is a professor in Lehigh, s is an
student in Lehigh and s is taking a class with
p
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Functions
Definition A function f from a subset A to a set
B, denoted by f A ? B, is a relation such that
for each a ?A there is one and only one b ?B
such that (a,b) ?f
No
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Functions (2)
Example of other functions in real life?
SSN People ? Social Security Number
The KEY Question When should we use functions
and when should we use relations?

• The KEY Answer
• We use functions if each element in A has to be
  related to one and only one element in B (think
  SSN)
• Otherwise, we use relations (think
  professor-student relation)

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Homework (for next class)

• Book
• 1.2 a)
• 1.4 c)
• 1.5
• True or false (explain your answers)
• A ? B B ? A
• A ? (B ? C) (A ? B) ? C
• Give an example of a real-life function (what
  is A? what is B?)
• Give an example of a related real-life relation
  (what is A? what is B?)

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Equivalence Relations
A relation R is an equivalence relations if R is
reflexive, R is symmetric and R is transitive
R is reflexive if (a,a) ? R for each a in the
language R is symmetric if the following property
holds if (a,b) ? R holds then (b,a) ? R also
holds R is transitive if the following property
holds (a,b) ? R and (b,c) ? R hold then (a,c) ?
R also hold
Equivalence relations are generalizations of the
equality relation
the relation x lt y