Predicates

... P(n) is computable On input any natural number given in unary describe a TM that decides whether this number is zero (i.e the empty string) or not. • High level program: – If the tape is empty accept else reject ...

... P(n) is computable On input any natural number given in unary describe a TM that decides whether this number is zero (i.e the empty string) or not. • High level program: – If the tape is empty accept else reject ...

(pdf)

... Because of this, it makes sense to restrict our questions about the computability of a function to functions from rational numbers to rational numbers. Because of the bijection between natural numbers and rationals, it also makes sense to further restrict the question to functions from the natural n ...

... Because of this, it makes sense to restrict our questions about the computability of a function to functions from rational numbers to rational numbers. Because of the bijection between natural numbers and rationals, it also makes sense to further restrict the question to functions from the natural n ...

on Computability

... rules, can be computed by one of TM. Consistent • Briefly, a Turing machine can be thought of as a black box, which performs a calculation of some kind on an input number. If the calculation reaches a conclusion, or halts then an output number is returned. • One of the consequences of Turing's theor ...

... rules, can be computed by one of TM. Consistent • Briefly, a Turing machine can be thought of as a black box, which performs a calculation of some kind on an input number. If the calculation reaches a conclusion, or halts then an output number is returned. • One of the consequences of Turing's theor ...

3-8 Unions and Intersection of Sets

... The Union of two or more sets is the set that contains all elements of the sets Symbol: ∪ How to find: list the elements that are in either set, or in both sets ...

... The Union of two or more sets is the set that contains all elements of the sets Symbol: ∪ How to find: list the elements that are in either set, or in both sets ...

Lecture 4

... The above properties are easy to prove. Thus, is an equivalence relation on the class of all sets. ...

... The above properties are easy to prove. Thus, is an equivalence relation on the class of all sets. ...

Recursive Enumerable

... If HALT is r.e., then Accept(M0) = HALT for some M0 . We’ll want a machine that either accepts or loops forever. So, we define a new machine, M, from M0 as follows. On any input: M behaves just like M0 until M0 stops (if it does). If M0 accepts, then M accepts. If M0 rejects, then M loops forever. ...

... If HALT is r.e., then Accept(M0) = HALT for some M0 . We’ll want a machine that either accepts or loops forever. So, we define a new machine, M, from M0 as follows. On any input: M behaves just like M0 until M0 stops (if it does). If M0 accepts, then M accepts. If M0 rejects, then M loops forever. ...

WEEK 1: CARDINAL NUMBERS 1. Finite sets 1.1. For a finite set A

... 1.2. If A, B are two finite sets then #(A × B) = #A#̇B. 1.3. P (A), the power set of A, has 2#A elements. To see this, assume A = {1, 2, .., n}. Attach to each subset S ⊂ A the binary string y = y1 y2 ...yn where yi = 1 if i ∈ S and 0 otherwise. Such a correspondence is bijective, and the number of ...

... 1.2. If A, B are two finite sets then #(A × B) = #A#̇B. 1.3. P (A), the power set of A, has 2#A elements. To see this, assume A = {1, 2, .., n}. Attach to each subset S ⊂ A the binary string y = y1 y2 ...yn where yi = 1 if i ∈ S and 0 otherwise. Such a correspondence is bijective, and the number of ...

STD VII - Kerala Samajam Model School

... Identify finite, infinite, empty and the singleton sets among the following: a. Set of odd numbers divisible by 2 b. Set of even prime numbers c. { x: x is right angle in an acute angled triangle} d. The set of positive integers greater than 100 e. { x:x is an obtuse angle in a scalene triangle} f. ...

... Identify finite, infinite, empty and the singleton sets among the following: a. Set of odd numbers divisible by 2 b. Set of even prime numbers c. { x: x is right angle in an acute angled triangle} d. The set of positive integers greater than 100 e. { x:x is an obtuse angle in a scalene triangle} f. ...

Basics of Sets

... So far in the worksheet, you have seen 8 sets: A, B, C, D, E, F, G and H. Write down the cardinality of each of these sets, using proper mathematical notation to express your answer. Do you think the infinite sets (if there are any) satisfy Cantor’s definition of countably infinite? Write down any r ...

... So far in the worksheet, you have seen 8 sets: A, B, C, D, E, F, G and H. Write down the cardinality of each of these sets, using proper mathematical notation to express your answer. Do you think the infinite sets (if there are any) satisfy Cantor’s definition of countably infinite? Write down any r ...

Document

... do we care? • Are any of these functions, ones that we would actually want to compute? – The argument does not even give any example of something that can’t be done, it just says that such an example exists ...

... do we care? • Are any of these functions, ones that we would actually want to compute? – The argument does not even give any example of something that can’t be done, it just says that such an example exists ...

2.1 Practice Using Set Notation HW

... Worksheet on equal sets and equivalent sets will help us to practice different types of questions to state whether the pairs of sets are equal sets or equivalent sets. We know, two sets are equal when they have same elements and two sets are equivalent when they have same number of elements whether ...

... Worksheet on equal sets and equivalent sets will help us to practice different types of questions to state whether the pairs of sets are equal sets or equivalent sets. We know, two sets are equal when they have same elements and two sets are equivalent when they have same number of elements whether ...

The Arithmetical Hierarchy Math 503

... effectively, i.e. by the partial recursive functions. These are the recursively enumerable sets (and as a special case of these, the recursive sets). These sets can be very limiting, however, and recursion theory—despite its profound importance in mathematical logic and computer science—may seem to ...

... effectively, i.e. by the partial recursive functions. These are the recursively enumerable sets (and as a special case of these, the recursive sets). These sets can be very limiting, however, and recursion theory—despite its profound importance in mathematical logic and computer science—may seem to ...

PDF

... then for k = 4 there would be the solution 2, 3, 7, 43. The verification would show that 3 · 7 · 43 + 1 = 904, and 2|904 2 · 7 · 43 + 1 = 603, and 3|603 2 · 3 · 43 + 1 = 259, and 7|259 2 · 3 · 7 + 1 = 43, and obviously 43|43 because 43 = 43. The vast majority of the sets in the known solutions inclu ...

... then for k = 4 there would be the solution 2, 3, 7, 43. The verification would show that 3 · 7 · 43 + 1 = 904, and 2|904 2 · 7 · 43 + 1 = 603, and 3|603 2 · 3 · 43 + 1 = 259, and 7|259 2 · 3 · 7 + 1 = 43, and obviously 43|43 because 43 = 43. The vast majority of the sets in the known solutions inclu ...

310409-Theory of computation

... • It is essential to have a criterion for determining, for any given thing, whether it is or is not a member of the given set. • This criterion is called the membership criterion of the set. ...

... • It is essential to have a criterion for determining, for any given thing, whether it is or is not a member of the given set. • This criterion is called the membership criterion of the set. ...

Document

... 7. (20) True or False. If the answer is true, provide an example (Hint: use subsets of integers and real numbers) as a proof. (1) The intersection of two countably infinite sets can be finite. ...

... 7. (20) True or False. If the answer is true, provide an example (Hint: use subsets of integers and real numbers) as a proof. (1) The intersection of two countably infinite sets can be finite. ...

MTH 231 - Shelton State

... 1. A classroom with 15 students and 15 textbooks. 2. A classroom with 16 students and 15 books. 3. A classroom with 15 students and 16 books. The implication of one-to-one is that the two sets under discussion have the same number of elements (more on this later). ...

... 1. A classroom with 15 students and 15 textbooks. 2. A classroom with 16 students and 15 books. 3. A classroom with 15 students and 16 books. The implication of one-to-one is that the two sets under discussion have the same number of elements (more on this later). ...

Word - Hostos Community College

... 1. Classify numbers as ordinal, or cardinal 2. Construct a one-to-one correspondence between the elements of two sets ...

... 1. Classify numbers as ordinal, or cardinal 2. Construct a one-to-one correspondence between the elements of two sets ...

HOSTOS COMMUNITY COLLEGE DEPARTMENT OF MATHEMATICS MAT 100

... 5. Evaluate solutions to problems for reasonableness. Recognize patterns and use these patterns for predicting the general term in a sequence. ...

... 5. Evaluate solutions to problems for reasonableness. Recognize patterns and use these patterns for predicting the general term in a sequence. ...

Solutions - Math Berkeley

... Solution: If we have two countable sets, then we can match one of them up with the even whole numbers {0, 2, 4, 6, . . .} and match the other up with the odd whole numbers {1, 3, 5, . . .}. Then the two sets together are matched up with the whole numbers {0, 1, 2, 3, . . .}, which shows that the two ...

... Solution: If we have two countable sets, then we can match one of them up with the even whole numbers {0, 2, 4, 6, . . .} and match the other up with the odd whole numbers {1, 3, 5, . . .}. Then the two sets together are matched up with the whole numbers {0, 1, 2, 3, . . .}, which shows that the two ...