Discrete Mathematics - Lyle School of Engineering
... (whom they prefer to their assigned mate) that is willing to elope with them. ...
... (whom they prefer to their assigned mate) that is willing to elope with them. ...
Methods of Proof for Boolean Logic
... 1. In giving an informal proof from some premises, if Q is already known to be a logical consequence of some already proven sentences, then you may assert Q in your proof. 2. Each step should be significant and easily understood (this is where ...
... 1. In giving an informal proof from some premises, if Q is already known to be a logical consequence of some already proven sentences, then you may assert Q in your proof. 2. Each step should be significant and easily understood (this is where ...
Peano`s Arithmetic
... back to the publisher with corrections and his own suggestions for improvement. He even asked for permission to publish Genocchi’s lectures. After fixing some errors and adding his own comments to the collection, he only listed himself as an editor [2]. In 1884 Peano became a professor at the univer ...
... back to the publisher with corrections and his own suggestions for improvement. He even asked for permission to publish Genocchi’s lectures. After fixing some errors and adding his own comments to the collection, he only listed himself as an editor [2]. In 1884 Peano became a professor at the univer ...
1. Sets, relations and functions. 1.1 Set theory. We assume the
... (ii.) If A is a partition of X and no member of A is empty then r = ∪{A × A : A ∈ A} is an equivalence relation on X and X/r = A. Proof. We leave this as Exercise 1.4. for the reader. Definition. Suppose X is a set, r is a relation on X and A ⊂ X. We say a member u of X is an upper bound for A if (a ...
... (ii.) If A is a partition of X and no member of A is empty then r = ∪{A × A : A ∈ A} is an equivalence relation on X and X/r = A. Proof. We leave this as Exercise 1.4. for the reader. Definition. Suppose X is a set, r is a relation on X and A ⊂ X. We say a member u of X is an upper bound for A if (a ...
ordinal logics and the characterization of informal concepts of proof
... is also a proof predicate, but Gonx8 is provable in (S) itself. To give a precise treatment of this idea of recognizing a proof predicate as such we shall consider formal systems whose constants are not only numerical terms and function symbols, but also proof predicates. This is independently justi ...
... is also a proof predicate, but Gonx8 is provable in (S) itself. To give a precise treatment of this idea of recognizing a proof predicate as such we shall consider formal systems whose constants are not only numerical terms and function symbols, but also proof predicates. This is independently justi ...
The superjump in Martin-Löf type theory
... Universes of types were introduced into constructive type theory by MartinLöf [11]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then ’reflects’ C. Several gadgets for generating u ...
... Universes of types were introduced into constructive type theory by MartinLöf [11]. The idea of forming universes in type theory is to introduce a universe as a set closed under a certain specified ensemble of set constructors, say C. The universe then ’reflects’ C. Several gadgets for generating u ...
Incompleteness - the UNC Department of Computer Science
... axioms but can be proven to be true in the larger system of set theory. Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. Goodstein's theorem is a relatively simple statement about natural numbers that is ...
... axioms but can be proven to be true in the larger system of set theory. Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. Goodstein's theorem is a relatively simple statement about natural numbers that is ...
MATHEMATICAL LOGIC CLASS NOTE 1. Propositional logic A
... (1) α is not satisfiable iff ¬α is tautology. (But a satisfiable sentence need not be tautology.) (2) α, β are tautologically equivalent iff α ↔ β is a tautology. (3) De Morgan’s law: ¬(φ ∧ ψ) ↔ ¬φ ∨ ¬ψ; ¬(φ ∨ ψ) ↔ ¬φ ∧ ¬ψ are both tautology. A literal is a sentence which is a sentence symbol, or the ...
... (1) α is not satisfiable iff ¬α is tautology. (But a satisfiable sentence need not be tautology.) (2) α, β are tautologically equivalent iff α ↔ β is a tautology. (3) De Morgan’s law: ¬(φ ∧ ψ) ↔ ¬φ ∨ ¬ψ; ¬(φ ∨ ψ) ↔ ¬φ ∧ ¬ψ are both tautology. A literal is a sentence which is a sentence symbol, or the ...
IS IT EASY TO LEARN THE LOGIC
... it is not generally emphasized the importance of truth tables as the foundation of the basic rules of elementary logic, because logical operations in the process of formulas transformation are artificial and mechanical. Likewise the use of logical rules in general, seems to be more involved in the c ...
... it is not generally emphasized the importance of truth tables as the foundation of the basic rules of elementary logic, because logical operations in the process of formulas transformation are artificial and mechanical. Likewise the use of logical rules in general, seems to be more involved in the c ...
santhanam_ratlocc2011
... • PRGs exist non-constructively for any natural class of properties (by the probabilistic method), but when do quick PRGs exist? • Theorem [NW, IW, KvM] : There is a quick PRG against SIZEA(poly) iff there are explicit functions which are hard against A-oracle circuits of size 2o(n) ...
... • PRGs exist non-constructively for any natural class of properties (by the probabilistic method), but when do quick PRGs exist? • Theorem [NW, IW, KvM] : There is a quick PRG against SIZEA(poly) iff there are explicit functions which are hard against A-oracle circuits of size 2o(n) ...
PPT
... Logic Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and formalized in the 19th and early 20th century. George Boole (Boolean algebra) introduced Math ...
... Logic Logic is not only the foundation of mathematics, but also is important in numerous fields including law, medicine, and science. Although the study of logic originated in antiquity, it was rebuilt and formalized in the 19th and early 20th century. George Boole (Boolean algebra) introduced Math ...
Propositional Logic
... Follow along in class rather than take notes Ask questions in class Keep up with the class Read the book, not just the slides ...
... Follow along in class rather than take notes Ask questions in class Keep up with the class Read the book, not just the slides ...
Midterm Exam 2 Solutions, Comments, and Feedback
... Problem 5: 15 points for the proof (part (a)) and 5 for the counterexample (part (b)). For exam statistics and a letter grade correspondence see the course webpage. • Solutions: Solutions, along with some remarks about common errors, are attached. Check the solutions first before asking questions ab ...
... Problem 5: 15 points for the proof (part (a)) and 5 for the counterexample (part (b)). For exam statistics and a letter grade correspondence see the course webpage. • Solutions: Solutions, along with some remarks about common errors, are attached. Check the solutions first before asking questions ab ...
ch1_Logic_and_proofs
... are definitions: Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles. Two angles are supplementary if the sum of their measures is 180 degrees. ...
... are definitions: Two triangles are congruent if their vertices can be paired so that the corresponding sides are equal and so are the corresponding angles. Two angles are supplementary if the sum of their measures is 180 degrees. ...
Upper-Bounding Proof Length with the Busy
... on the length used to encode the statement. This bound uses the Busy Beaver function, as shown in the next section, in which we reproduce Chaitin’s argument. This paper presents an analogous result about proofs. It is likewise commonly held that no matter how hard you’ve tried and failed to prove a ...
... on the length used to encode the statement. This bound uses the Busy Beaver function, as shown in the next section, in which we reproduce Chaitin’s argument. This paper presents an analogous result about proofs. It is likewise commonly held that no matter how hard you’ve tried and failed to prove a ...
If T is a consistent theory in the language of arithmetic, we say a set
... if a prime divides a product it divides one of its factors, and that if two numbers with no common prime factor both divide a number, then so does their product. (The reader may recognize these as results we took for granted in the proof of Lemma 16.5.) Once we have enough elementary lemmas, we can ...
... if a prime divides a product it divides one of its factors, and that if two numbers with no common prime factor both divide a number, then so does their product. (The reader may recognize these as results we took for granted in the proof of Lemma 16.5.) Once we have enough elementary lemmas, we can ...
Subintuitionistic Logics with Kripke Semantics
... In 1981, A. Visser [7] had already introduced Basic logic (BPC), an extension of F with truth preservation, in the natural deduction form, and proved completeness of BPC for finite, transitive, irreflexive Kripke models. Then in 1997, Suzuki and Ono [6] introduced a Hilbert style proof system for BP ...
... In 1981, A. Visser [7] had already introduced Basic logic (BPC), an extension of F with truth preservation, in the natural deduction form, and proved completeness of BPC for finite, transitive, irreflexive Kripke models. Then in 1997, Suzuki and Ono [6] introduced a Hilbert style proof system for BP ...
Methods of Proofs Recall we discussed the following methods of
... A proof is a sequence of statements bound together by the rules of logic, definitions, previously proven theorems, simple algebra and axioms. Definition: An integer n is even if there exists an integer k such that n = 2k. An integer n is odd if there exists an integer k such that n = 2k + 1. Example ...
... A proof is a sequence of statements bound together by the rules of logic, definitions, previously proven theorems, simple algebra and axioms. Definition: An integer n is even if there exists an integer k such that n = 2k. An integer n is odd if there exists an integer k such that n = 2k + 1. Example ...
3463: Mathematical Logic
... is applied to any configuration of the form αpaβ, or possibly αp if a is the blank symbol, and yields αbqβ. There are a few more cases to be considered for quintuples pabLq, but it is all quite simple. (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there ...
... is applied to any configuration of the form αpaβ, or possibly αp if a is the blank symbol, and yields αbqβ. There are a few more cases to be considered for quintuples pabLq, but it is all quite simple. (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there ...
An Introduction to Löb`s Theorem in MIRI Research
... (Here N is a parameter that doesn’t depend on X; we’ll think of it as some extremely large number. The only reason we have that parameter at all is so that our algorithm does in fact always return an output in finite time.) Some things are clear from the definition of FairBot. One is that it is capa ...
... (Here N is a parameter that doesn’t depend on X; we’ll think of it as some extremely large number. The only reason we have that parameter at all is so that our algorithm does in fact always return an output in finite time.) Some things are clear from the definition of FairBot. One is that it is capa ...
Philosophy as Logical Analysis of Science: Carnap, Schlick, Gödel
... truth and designation. But this error is easily corrected. If our ordinary notions of truth and designation are legitimate and nonparadoxical, they can be used in Tarski-style rules to state truth and designation conditions that provide some information about meaning. The requirement that for each s ...
... truth and designation. But this error is easily corrected. If our ordinary notions of truth and designation are legitimate and nonparadoxical, they can be used in Tarski-style rules to state truth and designation conditions that provide some information about meaning. The requirement that for each s ...
The Fundamental Theorem of World Theory
... Note that, given Coherence and some basic modal and propositional logic, the Equivalence Principle is equivalent to: The Leibniz Principle It is necessary that p if and only if p is true at every possible world. More formally, in terms of the language at hand: LP p ↔ ∀w(w |= p) Given this equivalen ...
... Note that, given Coherence and some basic modal and propositional logic, the Equivalence Principle is equivalent to: The Leibniz Principle It is necessary that p if and only if p is true at every possible world. More formally, in terms of the language at hand: LP p ↔ ∀w(w |= p) Given this equivalen ...