Preservation and destruction in simple refinements
... direction, let U be basic open neighborhood of x in hσ, Ai (that is, “basic” as determined by the subbasis σ ∪ {A}). First suppose that there is some V ⊆ U such that x ∈ V ∈ σ. Because σ is completely regular, there is a function f : X → [0, 1] that is 1 on {x} and 0 on X \ V , and f is continuous w ...
... direction, let U be basic open neighborhood of x in hσ, Ai (that is, “basic” as determined by the subbasis σ ∪ {A}). First suppose that there is some V ⊆ U such that x ∈ V ∈ σ. Because σ is completely regular, there is a function f : X → [0, 1] that is 1 on {x} and 0 on X \ V , and f is continuous w ...
Lecture 2
... • A proposition can be interpreted as being either true or false. For example: • “Henry VIII had one son and Cleopatra had two” • We wish to translate English propositions to Boolean expressions because: – English is ambiguous, computers require logical clarity. – We can automate, analyse, reason ab ...
... • A proposition can be interpreted as being either true or false. For example: • “Henry VIII had one son and Cleopatra had two” • We wish to translate English propositions to Boolean expressions because: – English is ambiguous, computers require logical clarity. – We can automate, analyse, reason ab ...
pdf
... some greater generality, it does not seem necessary for most applications. Agotnes and Alechina (2007) consider a restricted version of the HR logic, where we can reason only about certain types of unawareness (specifically, the only types of quantified statements allowed are ones that talk about wh ...
... some greater generality, it does not seem necessary for most applications. Agotnes and Alechina (2007) consider a restricted version of the HR logic, where we can reason only about certain types of unawareness (specifically, the only types of quantified statements allowed are ones that talk about wh ...
Introduction to Computational Logic
... This time the claim involves a boolean variable x and the proof proceeds by case analysis on x. Since reflexivity performs simplification automatically, we have omitted the tactic simpl. It is important that with Coq you step back and forth in the proof script and observe what happens. This way you ...
... This time the claim involves a boolean variable x and the proof proceeds by case analysis on x. Since reflexivity performs simplification automatically, we have omitted the tactic simpl. It is important that with Coq you step back and forth in the proof script and observe what happens. This way you ...
On two problems with the Theory of the Creating Subject
... Subject can register not only what objects it has created so far but also how and when, and that this reflection can be exploited to demonstrate mathematical theorems. An example is Brouwer’s argument published in [10] to the conclusion that we have at present no evidence for ∀x ∈ R(x ≠ 0 → x # 0) ...
... Subject can register not only what objects it has created so far but also how and when, and that this reflection can be exploited to demonstrate mathematical theorems. An example is Brouwer’s argument published in [10] to the conclusion that we have at present no evidence for ∀x ∈ R(x ≠ 0 → x # 0) ...
Euclid(A,B)
... Create GCD by multiplying together each common prime raised to the highest power that goes into both A and B. Give an algorithm to compute GCD(A,B) that does not require factoring A and B into primes, and does not simply try dividing by most numbers smaller than A and B to find the GCD. Run your alg ...
... Create GCD by multiplying together each common prime raised to the highest power that goes into both A and B. Give an algorithm to compute GCD(A,B) that does not require factoring A and B into primes, and does not simply try dividing by most numbers smaller than A and B to find the GCD. Run your alg ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.