MAA245 NUMBERS 1 Natural Numbers, N
... ∗, . . .; i.e. if, given a1 , a2 ∈ A and b1 , b2 ∈ B, with a1 ↔ b1 and a2 ↔ b2 , then a1 ∗ a2 ↔ b1 ∗ b2 , . . .. Thus hZ+ , +, ×i is isomorphic to hN, +, ×i under the bijection Ks ↔ s. Where convenient, we can denote the positive integer Ks = {(k + s, k); k ∈ N} by s. Also, denote the negative inte ...
... ∗, . . .; i.e. if, given a1 , a2 ∈ A and b1 , b2 ∈ B, with a1 ↔ b1 and a2 ↔ b2 , then a1 ∗ a2 ↔ b1 ∗ b2 , . . .. Thus hZ+ , +, ×i is isomorphic to hN, +, ×i under the bijection Ks ↔ s. Where convenient, we can denote the positive integer Ks = {(k + s, k); k ∈ N} by s. Also, denote the negative inte ...
Model theory makes formulas large
... there are first-order formulas ϕ for which the shortest equivalent formula in Gaifman normal form is non-elementarily larger than ϕ. Theorem 4.1. For every h ≥ 1 there is an FO(E)-sentence ϕh of size O(h4 ) such that every FO(E)-sentence in Gaifman normal form that is equivalent to ϕh on the class T ...
... there are first-order formulas ϕ for which the shortest equivalent formula in Gaifman normal form is non-elementarily larger than ϕ. Theorem 4.1. For every h ≥ 1 there is an FO(E)-sentence ϕh of size O(h4 ) such that every FO(E)-sentence in Gaifman normal form that is equivalent to ϕh on the class T ...
Factoring out the impossibility of logical aggregation
... deductively closed in the same relative sense, i.e., for all ∈ ∗ , if B, then ∈ B. It follows in particular that ∈ B ⇔ ∈ B when ↔ , and that ∈ B when . Deductive closure and its consequences raise the “logical omniscience problem” that is widely discussed in epistemic logic (see, ...
... deductively closed in the same relative sense, i.e., for all ∈ ∗ , if B, then ∈ B. It follows in particular that ∈ B ⇔ ∈ B when ↔ , and that ∈ B when . Deductive closure and its consequences raise the “logical omniscience problem” that is widely discussed in epistemic logic (see, ...
Fermat`s Little Theorem
... This process can be continued indefinitely to prove the result. (Technically, the result ap ≡ a mod p is found by induction on a.) An important use of this result is the following: Theorem: If a is not divisible by p, the inverse of a mod p is ap−2 . This is clearly true since 1 ≡ ap−1 ≡ a · ap−2 mo ...
... This process can be continued indefinitely to prove the result. (Technically, the result ap ≡ a mod p is found by induction on a.) An important use of this result is the following: Theorem: If a is not divisible by p, the inverse of a mod p is ap−2 . This is clearly true since 1 ≡ ap−1 ≡ a · ap−2 mo ...
EGYPTIAN FRACTIONS WITH EACH DENOMINATOR HAVING
... The existence of Egyptian fractions for any rational number has been known since at least Fibonacci (for example, the greedy algorithm will always produce a solution, though other methods are known). However, one can place additional constraints on the allowable ai and then interesting questions ari ...
... The existence of Egyptian fractions for any rational number has been known since at least Fibonacci (for example, the greedy algorithm will always produce a solution, though other methods are known). However, one can place additional constraints on the allowable ai and then interesting questions ari ...
CCGPS Advanced Algebra
... MCC9‐12.A.SSE.1bInterpret complicated expressions by viewing one or more of their parts as a single entity★ ...
... MCC9‐12.A.SSE.1bInterpret complicated expressions by viewing one or more of their parts as a single entity★ ...
CS 486: Applied Logic 8 Compactness (Lindenbaum`s Theorem)
... Theorem 8.7 (Deduction Theorem) If X is true under all interpretations that satisfy S then X is deducible from S. This means that a logical consequence of an infinite set of formulas can be proven on the basis of a finite subset of that set. Proof: Let X is true under all interpretations that satisf ...
... Theorem 8.7 (Deduction Theorem) If X is true under all interpretations that satisfy S then X is deducible from S. This means that a logical consequence of an infinite set of formulas can be proven on the basis of a finite subset of that set. Proof: Let X is true under all interpretations that satisf ...
Ramsey Theory
... Note that if an n0 has the required property, all bigger numbers have it as well. The smallest such n0 number is defined as the Ramsey number Rkr (a1 , ..., ak ). Also, |Y | ≥ ai can be changed to |Y | = ai with no effect on n0 . Furthermore, ”there exists an n0 positive integer, such that for all n ...
... Note that if an n0 has the required property, all bigger numbers have it as well. The smallest such n0 number is defined as the Ramsey number Rkr (a1 , ..., ak ). Also, |Y | ≥ ai can be changed to |Y | = ai with no effect on n0 . Furthermore, ”there exists an n0 positive integer, such that for all n ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.