Proofs in Propositional Logic
... In Coq, the negation of a proposition A is represented with the help of a constant not, where not A (also written ∼A) is defined as the implication A→False. The tactic unfold not allows to expand the constant not in a goal, but is seldom used. The introduction tactic for ∼A is the introduction tactic ...
... In Coq, the negation of a proposition A is represented with the help of a constant not, where not A (also written ∼A) is defined as the implication A→False. The tactic unfold not allows to expand the constant not in a goal, but is seldom used. The introduction tactic for ∼A is the introduction tactic ...
A54 INTEGERS 10 (2010), 733-745 REPRESENTATION NUMBERS
... k = 0, and p = 139 and k = 0. In Table 2 we list all the instances we found, for 11 ≤ p ≤ 89, for which the test of Theorem 17.17 worked. Corollary 18. If Mn = 2k+1 p for some k ≥ 0 and prime p, 11 ≤ p ≤ 5693, then rep(K1,n ) = Mn . ...
... k = 0, and p = 139 and k = 0. In Table 2 we list all the instances we found, for 11 ≤ p ≤ 89, for which the test of Theorem 17.17 worked. Corollary 18. If Mn = 2k+1 p for some k ≥ 0 and prime p, 11 ≤ p ≤ 5693, then rep(K1,n ) = Mn . ...
SMOOTH CONVEX BODIES WITH PROPORTIONAL PROJECTION
... A convex body in Rn is a compact convex set with nonempty interior. If K is a convex body and L a linear subspace of Rn , then K|L is the orthogonal projection of K onto L. Let G(n, i) be the Grassmannian of all i-dimensional linear subspaces of Rn . A central question in the geometric tomography of ...
... A convex body in Rn is a compact convex set with nonempty interior. If K is a convex body and L a linear subspace of Rn , then K|L is the orthogonal projection of K onto L. Let G(n, i) be the Grassmannian of all i-dimensional linear subspaces of Rn . A central question in the geometric tomography of ...
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... This notion of p-morphism of situations is a variant of standard notions of p-morphism of frames and structures [3]. It is well known that if there is a p-morphism from one structure to another, then the two structures satisfy the same formulas. An analogous result holds for situations. Theorem 2.5: ...
... This notion of p-morphism of situations is a variant of standard notions of p-morphism of frames and structures [3]. It is well known that if there is a p-morphism from one structure to another, then the two structures satisfy the same formulas. An analogous result holds for situations. Theorem 2.5: ...
Some Early Analytic Number Theory
... We can imagine multiplying out this infinite product of binomials and adding up all the terms to get a power series, which must of course be the power series (3). To multiply out the infinite product, we take one term from each factor and multiply them together. The only way to get a constant term i ...
... We can imagine multiplying out this infinite product of binomials and adding up all the terms to get a power series, which must of course be the power series (3). To multiply out the infinite product, we take one term from each factor and multiply them together. The only way to get a constant term i ...
Formal systems of fuzzy logic and their fragments∗
... The logic BCK plus this axiom of prelinearity will be the starting point for us in this paper—we call this logic Fuzzy BCK logic (FBCK for short). This logic is obviously complete with respect to the BCK-chains and this is the rationale for the name “Fuzzy BCK”, as the authors believe that completen ...
... The logic BCK plus this axiom of prelinearity will be the starting point for us in this paper—we call this logic Fuzzy BCK logic (FBCK for short). This logic is obviously complete with respect to the BCK-chains and this is the rationale for the name “Fuzzy BCK”, as the authors believe that completen ...
admissible and derivable rules in intuitionistic logic
... Γ C ∨ D and ` ∧s(Γ) . The definition of admissibility and the disjunction property of intuitionistic logic lead to: ` s(C) or ` s(D) . Then by weakening we obtain: ` Γ → s(C) or ` Γ → s(D) . Hence the definition of an Γ-identity yields: ` Γ → C or ` Γ → D . Then (ii) follows from (i) with C = D. R ...
... Γ C ∨ D and ` ∧s(Γ) . The definition of admissibility and the disjunction property of intuitionistic logic lead to: ` s(C) or ` s(D) . Then by weakening we obtain: ` Γ → s(C) or ` Γ → s(D) . Hence the definition of an Γ-identity yields: ` Γ → C or ` Γ → D . Then (ii) follows from (i) with C = D. R ...
Grade 3 Math Flipchart
... Finds these statistical measures of a data set with less than ten data points using whole numbers from 0 through 1,000: a) minimum and maximum data values; b) range; c) mode (uni-modal only); d) median when data set has an odd number of data points Explanation of Indicator Using information students ...
... Finds these statistical measures of a data set with less than ten data points using whole numbers from 0 through 1,000: a) minimum and maximum data values; b) range; c) mode (uni-modal only); d) median when data set has an odd number of data points Explanation of Indicator Using information students ...
Notes on the History of Mathematics
... than talking about the area of a circle, the problem talks about a “round field”. There is little, if any, geometric abstraction in extant Babylonian and Egyptian texts. • We have no idea what a “khet” or a “setat” is, but we can infer it from context; one setat is presumably one square khet. In par ...
... than talking about the area of a circle, the problem talks about a “round field”. There is little, if any, geometric abstraction in extant Babylonian and Egyptian texts. • We have no idea what a “khet” or a “setat” is, but we can infer it from context; one setat is presumably one square khet. In par ...
John L. Pollock
... name). Initially, the empty set is apt to seem paradoxical. How can we have a collection without anything in it? But reflection shows that the empty set is not really so strange as it may first appear. For example, a mathematician might consider the set of all solutions to a particular equation, pro ...
... name). Initially, the empty set is apt to seem paradoxical. How can we have a collection without anything in it? But reflection shows that the empty set is not really so strange as it may first appear. For example, a mathematician might consider the set of all solutions to a particular equation, pro ...
A007970: Proof of a Theorem Related to the Happy Number
... . D and E are both odd if d is odd (in fact ≡ 3 (mod 4)), and they are U0 both even if d is even (in fact ≡ 0 (mod 4)). The solutions U0 , T0 are the minimal positive ones. Proof: One uses the basic result that the Pell eq. (2) has a fundamental positive solution (x0 , y0 ) for each non-square posit ...
... . D and E are both odd if d is odd (in fact ≡ 3 (mod 4)), and they are U0 both even if d is even (in fact ≡ 0 (mod 4)). The solutions U0 , T0 are the minimal positive ones. Proof: One uses the basic result that the Pell eq. (2) has a fundamental positive solution (x0 , y0 ) for each non-square posit ...
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.