Teach Yourself Logic 2016: A Study Guide
... L3. If you have taken an elementary logic course based on a substantial text like the ones mentioned in this section, then you should be well prepared. Here then, for those that need them, are two initial suggestions of formal logic books that start from scratch and go far enough to provide a good f ...
... L3. If you have taken an elementary logic course based on a substantial text like the ones mentioned in this section, then you should be well prepared. Here then, for those that need them, are two initial suggestions of formal logic books that start from scratch and go far enough to provide a good f ...
A Transition to Advanced Mathematics
... statement is always true, so while the statement may be true for many (even infinitely many) examples, we would never know whether another example might show the statement to be false. By studying examples, we might conclude that the statement “x 2 − 3x + 43 is a prime number” is true for all positi ...
... statement is always true, so while the statement may be true for many (even infinitely many) examples, we would never know whether another example might show the statement to be false. By studying examples, we might conclude that the statement “x 2 − 3x + 43 is a prime number” is true for all positi ...
Modal Logic - Web Services Overview
... Sentence “ it is possible that this Boolean function is satisfied” is equivalent to the sentence “it is not necessary that this Boolean function is not satisfied”. ...
... Sentence “ it is possible that this Boolean function is satisfied” is equivalent to the sentence “it is not necessary that this Boolean function is not satisfied”. ...
X - UOW
... comprised of n statements, there will be 2n combinations of truth values. This method can be long for large numbers of statements. We will consider a quicker method for determining if a compound statement is a tautology. However, truth tables are reliable (“safe”) and are highly recommended if the “ ...
... comprised of n statements, there will be 2n combinations of truth values. This method can be long for large numbers of statements. We will consider a quicker method for determining if a compound statement is a tautology. However, truth tables are reliable (“safe”) and are highly recommended if the “ ...
On integers n for which X n – 1 has divisors of every degree
... This paper discusses the related concept of ϕ-practical numbers: A positive integer n is ϕ-practical if X n − 1 has divisors in Z[X] of every degree to n. Since X n − 1 is squarefree with its irreducible factors having degrees ϕ(d) as d runs over the divisors of n (where ϕ is Euler’s function), it f ...
... This paper discusses the related concept of ϕ-practical numbers: A positive integer n is ϕ-practical if X n − 1 has divisors in Z[X] of every degree to n. Since X n − 1 is squarefree with its irreducible factors having degrees ϕ(d) as d runs over the divisors of n (where ϕ is Euler’s function), it f ...
On the b-ary Expansion of an Algebraic Number.
... xi lie in the set f0; 1; . . . ; [b]g. We direct the reader to [4] and to the references quoted therein for more on b-expansions. We stress that the bexpansion of 1 has been extensively studied. It turns out that the method of proof of Theorem 1 applies to b-expansions, when b is a Pisot or a Salem ...
... xi lie in the set f0; 1; . . . ; [b]g. We direct the reader to [4] and to the references quoted therein for more on b-expansions. We stress that the bexpansion of 1 has been extensively studied. It turns out that the method of proof of Theorem 1 applies to b-expansions, when b is a Pisot or a Salem ...
The Premiss-Based Approach to Logical Aggregation Franz Dietrich & Philippe Mongin
... Imagine that a group of individuals needs to form collective ("yes" or "no") judgments on certain propositions, e.g., that smoking is harmful (a), smoking should be banned in public places (b), if smoking is harmful, then it should be banned in public places (a ! b), and so on. Assume further that t ...
... Imagine that a group of individuals needs to form collective ("yes" or "no") judgments on certain propositions, e.g., that smoking is harmful (a), smoking should be banned in public places (b), if smoking is harmful, then it should be banned in public places (a ! b), and so on. Assume further that t ...
25(4)
... that the second and fifth lines of the chain of equalities above are the same, by virtue of (2.4) and (2.5). Some interesting results for particular values of a and b may be found. For example, with a = 0, b = 2, we have, by (2,5) and (2.8), = 4 ^ ' 1 } (x) + 2«n(2) = 4(1 + x2)^'^ ...
... that the second and fifth lines of the chain of equalities above are the same, by virtue of (2.4) and (2.5). Some interesting results for particular values of a and b may be found. For example, with a = 0, b = 2, we have, by (2,5) and (2.8), = 4 ^ ' 1 } (x) + 2«n(2) = 4(1 + x2)^'^ ...
A DUAL APPROACH TO TRIANGLE SEQUENCES
... fractions can be defined via adding vectors to get as close as possible to a given ray without crossing the ray. Section 6 is the longest and most difficult of this paper. The goal of this section is to give a sharp description of precisely when a triangle sequence corresponds to a unique pair of numbe ...
... fractions can be defined via adding vectors to get as close as possible to a given ray without crossing the ray. Section 6 is the longest and most difficult of this paper. The goal of this section is to give a sharp description of precisely when a triangle sequence corresponds to a unique pair of numbe ...
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.