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Predicate logic. Formal and informal proofs
... Methods of proving theorems Basic methods to prove the theorems: • Direct proof – p q is proved by showing that if p is true then q follows • Indirect proof – Show the contrapositive ¬q ¬p. If ¬q holds then ¬p follows • Proof by contradiction – Show that (p ¬ q) contradicts the assumptions • ...
... Methods of proving theorems Basic methods to prove the theorems: • Direct proof – p q is proved by showing that if p is true then q follows • Indirect proof – Show the contrapositive ¬q ¬p. If ¬q holds then ¬p follows • Proof by contradiction – Show that (p ¬ q) contradicts the assumptions • ...
PPT
... with the aid of existing mathematical facts and theorems as well as laws of logic and reasoning, arrive at the conclusion Q. Although the goal is always to “go from P to Q,” is more than one valid way of achieving this goal. Five equivalent ways to proving the implication PQ are shown in next ...
... with the aid of existing mathematical facts and theorems as well as laws of logic and reasoning, arrive at the conclusion Q. Although the goal is always to “go from P to Q,” is more than one valid way of achieving this goal. Five equivalent ways to proving the implication PQ are shown in next ...
Full text
... from one of the Hermite theorems and it appeared evident that the criterion also held in the Fibonomial a r r a y , but the proof was not completed. In the present paper I remove all these defects by proving the Hermite theorems in a m o r e elegant manner so that very little needs to be assumed for ...
... from one of the Hermite theorems and it appeared evident that the criterion also held in the Fibonomial a r r a y , but the proof was not completed. In the present paper I remove all these defects by proving the Hermite theorems in a m o r e elegant manner so that very little needs to be assumed for ...
1 Proof by Contradiction - Stony Brook Mathematics
... We recall that strong induction was already used in the previous notes to show that we can write arbitrary sums without parentheses (a priori the associativity axiom only tells us this for sums of 3 terms). The reader is invited to revisit that proof. We now prove Theorem 4, that one of the players ...
... We recall that strong induction was already used in the previous notes to show that we can write arbitrary sums without parentheses (a priori the associativity axiom only tells us this for sums of 3 terms). The reader is invited to revisit that proof. We now prove Theorem 4, that one of the players ...
G¨ODEL`S COMPLETENESS AND INCOMPLETENESS
... 2.3. Implications of Completeness. This completeness theorem is really a statement about the relationship between languages and structures. The theorem itself states that if a set of formulas is consistent, then it is satisfiable. What this essentially means is that if a formula satisfies every stru ...
... 2.3. Implications of Completeness. This completeness theorem is really a statement about the relationship between languages and structures. The theorem itself states that if a set of formulas is consistent, then it is satisfiable. What this essentially means is that if a formula satisfies every stru ...
Proving Angle Relationships (Geom) WS pg. 99
... Supplementary and Complementary Angles There are two basic postulates for working with angles. The Protractor Postulate assigns numbers to angle measures, and the Angle Addition Postulate relates parts of an angle to the whole angle. Protractor Postulate ...
... Supplementary and Complementary Angles There are two basic postulates for working with angles. The Protractor Postulate assigns numbers to angle measures, and the Angle Addition Postulate relates parts of an angle to the whole angle. Protractor Postulate ...
(A B) |– A
... (tautologies). In this case the axioms have to be logically valid formulas (true under all interpretations), and the deduction rules have to make it possible to prove logically valid formulas. For that reason the rules are either truth-preserving or tautology preserving, i.e., A1,…,Am |– B1,…,Bm can ...
... (tautologies). In this case the axioms have to be logically valid formulas (true under all interpretations), and the deduction rules have to make it possible to prove logically valid formulas. For that reason the rules are either truth-preserving or tautology preserving, i.e., A1,…,Am |– B1,…,Bm can ...
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.