Herbrands Theorem
... Let , S {P ( x) Q( x), R ( f ( y ))} Herbrand Universe H H {a, f (a ), f ( f (a )), } Atom Set A is given by A {P (a ), P ( f (a )), P ( f ( f (a ))), , Q (a ), , R (a ), } Some Herbrand Interpreta tions are I 1 {P (a ), P ( f (a )), P ( f ( f (a ))), , Q (a ), , R (a ), ...
... Let , S {P ( x) Q( x), R ( f ( y ))} Herbrand Universe H H {a, f (a ), f ( f (a )), } Atom Set A is given by A {P (a ), P ( f (a )), P ( f ( f (a ))), , Q (a ), , R (a ), } Some Herbrand Interpreta tions are I 1 {P (a ), P ( f (a )), P ( f ( f (a ))), , Q (a ), , R (a ), ...
Ramsey`s Theorem and Compactness
... If f is an n-coloring of X, a subset Y ⊆ X is homogeneous or monochromatic for f if there is some i ∈ C such that, for every s ∈ Y [n] , we have f (s) = i. We call Y monochromatic because all n-element subsets of Y have the same color i. In the terms of this definition, the fact we proved in Section ...
... If f is an n-coloring of X, a subset Y ⊆ X is homogeneous or monochromatic for f if there is some i ∈ C such that, for every s ∈ Y [n] , we have f (s) = i. We call Y monochromatic because all n-element subsets of Y have the same color i. In the terms of this definition, the fact we proved in Section ...
Partial Correctness Specification
... A proof in Floyd-Hoare logic is a sequence of lines, each of which is either an axiom of the logic or follows from earlier lines by a rule of inference of the logic u ...
... A proof in Floyd-Hoare logic is a sequence of lines, each of which is either an axiom of the logic or follows from earlier lines by a rule of inference of the logic u ...
G - Courses
... terms according to the equalities between them in some structure satisfying the FO-sentence at hand. Here, we used the resolution procedure only for formulas of propositional logic. The resolution procedure can be extended to FO-formulas using unification of terms. There are other proofs of Göde ...
... terms according to the equalities between them in some structure satisfying the FO-sentence at hand. Here, we used the resolution procedure only for formulas of propositional logic. The resolution procedure can be extended to FO-formulas using unification of terms. There are other proofs of Göde ...
1.4 The set of Real Numbers: Quick Definition and
... We will not discuss those here. p Before we look at the properties of the real numbers, we prove that 2 is irrational using Pythagora’ proof. We begin with a lemma which proof is left as an exercise. Lemma 73 Let n be an integer. If n2 is even then n must also be even. Proof. See exercises. We are n ...
... We will not discuss those here. p Before we look at the properties of the real numbers, we prove that 2 is irrational using Pythagora’ proof. We begin with a lemma which proof is left as an exercise. Lemma 73 Let n be an integer. If n2 is even then n must also be even. Proof. See exercises. We are n ...
Class Notes
... In this example, the axioms (called premises or hypotheses) are written above the line and the theorem (called the conclusion) is written below the line. The whole argument is called a deduction. This particular argument is an example of a rule of inference which is now usually called Universal Inst ...
... In this example, the axioms (called premises or hypotheses) are written above the line and the theorem (called the conclusion) is written below the line. The whole argument is called a deduction. This particular argument is an example of a rule of inference which is now usually called Universal Inst ...
Special Products – Blue Level Problems In
... Algebraic conclusion: (n + 1)2 - n2 = (n + 1) + n note: Strictly you can use "n - 1" as a substitute for "n" and "n" as substitute for "n + 1". It won't make any changes to the final solution. "Banyak jalan menuju Roma" meaning: There are many ways to find an answer. b. The difference of every squar ...
... Algebraic conclusion: (n + 1)2 - n2 = (n + 1) + n note: Strictly you can use "n - 1" as a substitute for "n" and "n" as substitute for "n + 1". It won't make any changes to the final solution. "Banyak jalan menuju Roma" meaning: There are many ways to find an answer. b. The difference of every squar ...
Enhancing Your Subject Knowledge
... • Probably called a stadium because stadiums are shaped like it. • Strange how we don’t learn the name of this shape but it crops up in GCSE questions all the time! • Also called an obround and a discorectangle! ...
... • Probably called a stadium because stadiums are shaped like it. • Strange how we don’t learn the name of this shape but it crops up in GCSE questions all the time! • Also called an obround and a discorectangle! ...
Chapter 2 A Primer of Mathematical Writing (Proofs)
... Assume P Λ Q, deduce Try this approach when Q says a contradiction something is not true. Proof by Cases Break the domain into Try this for proving two or more subsets properties of numbers and prove PQ for the where odd and even elements in each such or positive and negative numbers subset. requir ...
... Assume P Λ Q, deduce Try this approach when Q says a contradiction something is not true. Proof by Cases Break the domain into Try this for proving two or more subsets properties of numbers and prove PQ for the where odd and even elements in each such or positive and negative numbers subset. requir ...
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.