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
Why Math? Math provides the tools to make these definitions unambiguous Among others, Today’s computers Again we need math to do this • 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
Why Math? (II) Finite Automata regular languages Pushdown Automata Context-free Languages Turing Machines Turing-computable languages We are going to define formal models of computation Name of Model Kind of language
Sets • Sets are denoted by { <collection of elements> } • 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”
Set Inclusion and Set Equality True or false: • {e,i,t,c} {a, b, …, z} • for any set A, A A • for any set A, A {A} 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
Cartesian Product pair or 2-tuple 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
Cartesian Product II Triple or 3-tuple 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
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)
Conventions and Precedence (1) (1) (2) (2) (1) ( (8 2) + (3 + 7)) A (B C) D
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}
Functions Question: Is the following relation: {(p,s) : p is a professor in Lehigh, s is an student in Lehigh and s is taking a class with p} a function from {p: professor in Lehigh} to {s: is an student in Lehigh}? 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
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)
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?)
Equivalence Relations {(1,2),(1,3),…, (2,3),(2,4),…} {(a,b) such that a and b are artificial lights of color red} 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 < y”
