1. 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

2. Why Math? We will formulate precise definitions of different kinds of automata We will formulate and proof properties of these automata Math provides the tools to make these definitions unambiguous Among others, Today’s computers Again we need math to do this Along the way we will formulate precisely: Nondeterminism Computation State (configuration) of a computation

3. Why Math? (II) We are going to define formal models of computation Name of ModelKind of language Finite Automata regular languages Pushdown AutomataContext-free Languages Turing MachinesTuring-computable languages

4. Sets Sets are denoted by { } 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”

5. 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 or false:  {e,i,t,c}  {a, b, …, z}  for any set A, A  A  for any set A, A  {A} true false Definition: Given 2 sets A and B, A is equal to B, denoted by A = B, if A  B and B  A

6. 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} true false pair or 2-tuple {(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}

7. 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: A 1  A 2  …  A n and equality between n-tuples Triple or 3-tuple

8. 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)

9. Conventions and Precedence ( (8  2) + (3 + 7)) A  (B  C)  D (1) (2) (1)

10. 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}

11. 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 {(p,s) : p is a professor in Lehigh, s is an student in Lehigh and s is taking a class with p} Question: Is the following relation: a function from {p: professor in Lehigh} to {s: is an student in Lehigh}? No

12. 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)

13. 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?)

14. 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 {(1,2),(1,3),…, (2,3),(2,4),…} {(a,b) such that a and b are artificial lights of color red} “the relation x < y”