Infinity and Diagonalization - Carnegie Mellon School of Computer
... Cantor’s definition only requires that some 1-1 correspondence between the two sets is onto, not that all 1-1 correspondences are onto. This distinction never arises when the sets are finite. ...
... Cantor’s definition only requires that some 1-1 correspondence between the two sets is onto, not that all 1-1 correspondences are onto. This distinction never arises when the sets are finite. ...
Something from Nothing
... The two questions are related. The A(x) stands for any valid formula in the language that involves the variable x, and for every possible such formula, there is another axiom. Thus, there are an infinite number of subset axioms. All the axioms have the same form, but different formulas replacing the ...
... The two questions are related. The A(x) stands for any valid formula in the language that involves the variable x, and for every possible such formula, there is another axiom. Thus, there are an infinite number of subset axioms. All the axioms have the same form, but different formulas replacing the ...
ON HIERARCHIES AND SYSTEMS OF NOTATIONS
... in B is recursive in this set (and if A is EL}in B, A is even many-one reducible to this set). This justifies calling the operation ...
... in B is recursive in this set (and if A is EL}in B, A is even many-one reducible to this set). This justifies calling the operation ...
The Fibonacci Numbers And An Unexpected Calculation.
... Problem from a 15-251 final: Show that the set of real numbers in [0,1] whose decimal expansion has the property that every digit is a prime number (2,3,5, or 7) is uncountable. E.g., 0.2375 and 0.55555… are in the set, but 0.145555… and 0.3030303… are not. ...
... Problem from a 15-251 final: Show that the set of real numbers in [0,1] whose decimal expansion has the property that every digit is a prime number (2,3,5, or 7) is uncountable. E.g., 0.2375 and 0.55555… are in the set, but 0.145555… and 0.3030303… are not. ...
MATH 2420 Discrete Mathematics
... set A collection of discrete items, whether numbers, letters, people, animals, cars, atoms, planets, etc. A set is identified when the items are grouped together between { and }. element One item in a set, whether a discrete item or a set itself. cardinality The number of elements in a finite set. S ...
... set A collection of discrete items, whether numbers, letters, people, animals, cars, atoms, planets, etc. A set is identified when the items are grouped together between { and }. element One item in a set, whether a discrete item or a set itself. cardinality The number of elements in a finite set. S ...
Lecture07_DecidabilityandDiagonalizationandCardinality
... If there exists a TM M that accepts every string in L, and loops for all strings not in L, then L is: a) Turing decidable b) Turing recognizable, but not decidable c) Not enough information ...
... If there exists a TM M that accepts every string in L, and loops for all strings not in L, then L is: a) Turing decidable b) Turing recognizable, but not decidable c) Not enough information ...
11 infinity
... Cantor’s definition only requires that some 1-1 correspondence between the two sets is onto, not that all 1-1 correspondences are onto. This distinction never arises when the sets are finite. ...
... Cantor’s definition only requires that some 1-1 correspondence between the two sets is onto, not that all 1-1 correspondences are onto. This distinction never arises when the sets are finite. ...
How To Think Like A Computer Scientist
... Cantor’s definition only requires that some 1-1 correspondence between the two sets is onto, not that all 1-1 correspondences are onto. This distinction never arises when the sets are finite. ...
... Cantor’s definition only requires that some 1-1 correspondence between the two sets is onto, not that all 1-1 correspondences are onto. This distinction never arises when the sets are finite. ...
Day 01 - Introduction to Sets
... A finite set has a __________________ number of elements and an infinite set has an infinite number of elements. The _____________ _________ is denoted as { } or . ...
... A finite set has a __________________ number of elements and an infinite set has an infinite number of elements. The _____________ _________ is denoted as { } or . ...
Chapter 1: Sets
... • Let A be any set and let B be any non-empty subset. Then A is called a proper subset of B, and is written as A B, if and only if every element of A is in B, and there exists at least one element in B which is not there in A. – i.e. if A ⊆ B and A B, then A B – Please note that ϕ has no prope ...
... • Let A be any set and let B be any non-empty subset. Then A is called a proper subset of B, and is written as A B, if and only if every element of A is in B, and there exists at least one element in B which is not there in A. – i.e. if A ⊆ B and A B, then A B – Please note that ϕ has no prope ...
Section 2.5
... Sets in One-to-One Correspondence with N We will see there are many different sets that are in one-to-one correspondnece with the set of natural numbers N. One of the most famous examples is the set of integers I. This seems very counter intuitive since there seems like there should be "twice" as m ...
... Sets in One-to-One Correspondence with N We will see there are many different sets that are in one-to-one correspondnece with the set of natural numbers N. One of the most famous examples is the set of integers I. This seems very counter intuitive since there seems like there should be "twice" as m ...
Chapter 1: Sets, Operations and Algebraic Language
... Note that the dot on the number line where the number 1 is located is open in the first example and closed in the second example. The open dot indicates that the number 1 is not an element of the set of all numbers greater than 1. The closed dot on the second number line indicates that 1 is an eleme ...
... Note that the dot on the number line where the number 1 is located is open in the first example and closed in the second example. The open dot indicates that the number 1 is not an element of the set of all numbers greater than 1. The closed dot on the second number line indicates that 1 is an eleme ...
Infinity and Diagonalization
... • Then h(x) = g(f(x)) is a bijection from AC • It follows that N, E, and Z • all have the same cardinality. ...
... • Then h(x) = g(f(x)) is a bijection from AC • It follows that N, E, and Z • all have the same cardinality. ...
COMPARING SETS Definition: EQUALITY OF 2 SETS Two sets A
... Ex: For the set {a, c, f}, there are Let’s list them: ...
... Ex: For the set {a, c, f}, there are Let’s list them: ...
The Foundations: Logic and Proofs
... Introduction • Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. – Important for counting. – Programming languages have set operations. • Set theory is an important branch of mathematics. – Many different systems of axioms have been used to devel ...
... Introduction • Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. – Important for counting. – Programming languages have set operations. • Set theory is an important branch of mathematics. – Many different systems of axioms have been used to devel ...
PDF
... (c) if g(n, s) 6= ∞ and g(n, s + 1) = ∞ then g(n, s) ⊂ S. We refer to g as a witness to f being S-limitwise monotonic. Note that if f is an S-limitwise monotonic function then its witness g can be chosen to be primitive recursive. 2.2 Definition. A collection of finite sets is S-monotonically appro ...
... (c) if g(n, s) 6= ∞ and g(n, s + 1) = ∞ then g(n, s) ⊂ S. We refer to g as a witness to f being S-limitwise monotonic. Note that if f is an S-limitwise monotonic function then its witness g can be chosen to be primitive recursive. 2.2 Definition. A collection of finite sets is S-monotonically appro ...
Problems set
... D. Recursively and iteratively show any linked list backwards. Discuss benefits and deficits of each approach. E. Recursively and iteratively show an array of any object backwards. Discuss benefits and deficits of each approach. F. Recursively and iteratively calculate compound Interest: input amoun ...
... D. Recursively and iteratively show any linked list backwards. Discuss benefits and deficits of each approach. E. Recursively and iteratively show an array of any object backwards. Discuss benefits and deficits of each approach. F. Recursively and iteratively calculate compound Interest: input amoun ...
Lesson 1 – Types of Sets and Set Notation
... Disjoint – Two or more sets having no elements in common Finite Set – A set with a countable number of elements Infinite Set – A set with an infinite number of elements ...
... Disjoint – Two or more sets having no elements in common Finite Set – A set with a countable number of elements Infinite Set – A set with an infinite number of elements ...
Chapter 1
... 2.1.2.3.2. Example – A = {, , } and B = {blue, gum, reading, playing}, then n(A) = 3 and n(B) = 4 2.1.2.4. The set of whole numbers is an infinite set designated in the following way: W = {0, 1, 2, 3, …} 2.1.2.5. Counting: the process that enables people systematically to associate a whole number ...
... 2.1.2.3.2. Example – A = {, , } and B = {blue, gum, reading, playing}, then n(A) = 3 and n(B) = 4 2.1.2.4. The set of whole numbers is an infinite set designated in the following way: W = {0, 1, 2, 3, …} 2.1.2.5. Counting: the process that enables people systematically to associate a whole number ...
If n = 1, then n!
... Exercises: pp. 328 ff. #4, 10, 15 Wednesday: Overview and discussion of final exam Evaluations ...
... Exercises: pp. 328 ff. #4, 10, 15 Wednesday: Overview and discussion of final exam Evaluations ...
Advanced Topics in Theoretical Computer Science
... Decidability and Undecidability results Theorem. It is undecidable whether a first order logic formula is valid. Proof. Suppose there is an algorithm P that, given a first order logic and a formula in that logic, decides whether that formula is valid. We use P to give a decision algorithm for the l ...
... Decidability and Undecidability results Theorem. It is undecidable whether a first order logic formula is valid. Proof. Suppose there is an algorithm P that, given a first order logic and a formula in that logic, decides whether that formula is valid. We use P to give a decision algorithm for the l ...
Lecture24 – Infinite sets
... Sets exist whose size is א0, א1, א2, א3… An infinite number of aleph numbers! ...
... Sets exist whose size is א0, א1, א2, א3… An infinite number of aleph numbers! ...
Annals of Pure and Applied Logic Ordinal machines and admissible
... second author recast the proof of the Sacks–Simpson theorem using the computational paradigm instead of constructibility theory. The crucial point involved was how the informally presented recursions in the argument of Sacks and Simpson [11, 12] (and a recursion method presented by Shore [13]) can b ...
... second author recast the proof of the Sacks–Simpson theorem using the computational paradigm instead of constructibility theory. The crucial point involved was how the informally presented recursions in the argument of Sacks and Simpson [11, 12] (and a recursion method presented by Shore [13]) can b ...