![Section 3.1: Direct Proof and Counterexample 1](http://s1.studyres.com/store/data/000451715_1-f57b8d2b4d5486c9e30b913acd4f81ad-300x300.png)
The Logic of Conditionals
... Completeness: If Q is a tautological consequence of P1,…,Pn, then P1,…,Pn -T Q. So, once you see that Q is a tautological consequence of P1,…,Pn, you can be sure that there is an FT-proof of Q from P1,…,Pn, even if you have not actually found such a proof. ...
... Completeness: If Q is a tautological consequence of P1,…,Pn, then P1,…,Pn -T Q. So, once you see that Q is a tautological consequence of P1,…,Pn, you can be sure that there is an FT-proof of Q from P1,…,Pn, even if you have not actually found such a proof. ...
SUMS OF DISTINCT UNIT FRACTIONS
... Herbert S. Wilf raises several questions about i?-bases, including: Does an i?-basis necessarily have a positive density? If 5 consists of all positive integers and /(») is the least number required to represent », what, in some average sense, is the growth of /(»)? These two questions are answered ...
... Herbert S. Wilf raises several questions about i?-bases, including: Does an i?-basis necessarily have a positive density? If 5 consists of all positive integers and /(») is the least number required to represent », what, in some average sense, is the growth of /(»)? These two questions are answered ...
3-6 Fundamental Theorem of Algebra Day 1
... as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers. ...
... as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers. ...
10 Inference
... whose square is larger than 25. The statement q is that n is larger than 5. We could argue directly but then we would need to know something about talking square roots. Instead, let us argue indirectly. Suppose ¬q, that is, n ≤ 5. By monotonicity of multiplication, we have ...
... whose square is larger than 25. The statement q is that n is larger than 5. We could argue directly but then we would need to know something about talking square roots. Instead, let us argue indirectly. Suppose ¬q, that is, n ≤ 5. By monotonicity of multiplication, we have ...
The strong law of large numbers - University of California, Berkeley
... others and deals with the case of identically distributed, independent random variables (r.v.'s). In this case it is known, after Kolmogorov,1 that a nasc for the validity of the SLLN is the finiteness of the first absolute moment of the common distribution function (d.f.). For use in certain statis ...
... others and deals with the case of identically distributed, independent random variables (r.v.'s). In this case it is known, after Kolmogorov,1 that a nasc for the validity of the SLLN is the finiteness of the first absolute moment of the common distribution function (d.f.). For use in certain statis ...
A(x)
... reasonable, for we couldn’t perform proofs if we did not know which formulas are axioms). It means that there is an algorithm that for any WFF given as its input answers in a finite number of steps an output Yes or NO on the question whether is an axiom or not. A finite set is trivially decidabl ...
... reasonable, for we couldn’t perform proofs if we did not know which formulas are axioms). It means that there is an algorithm that for any WFF given as its input answers in a finite number of steps an output Yes or NO on the question whether is an axiom or not. A finite set is trivially decidabl ...
Topic/ Theme/ Duration Pythagorean Theorem
... Solve one‐variable linear equations, including cases with infinitely many solutions or no solutions Solve problems involving volumes of cones, cylinders, and spheres together with previous geometry work, proportional reasoning and multi‐ step problem solving in grade 7 ...
... Solve one‐variable linear equations, including cases with infinitely many solutions or no solutions Solve problems involving volumes of cones, cylinders, and spheres together with previous geometry work, proportional reasoning and multi‐ step problem solving in grade 7 ...
A(x)
... if SA |= T then SA |– T (SA – a set of assumptions). But Hilbert wanted more: he supposed that all the truths of mathematics can be proved in this mechanic finite way. That is, that a theory of arithmetic (e.g. Peano) is complete in the following sense: each formula is in the theory decidable, i.e., ...
... if SA |= T then SA |– T (SA – a set of assumptions). But Hilbert wanted more: he supposed that all the truths of mathematics can be proved in this mechanic finite way. That is, that a theory of arithmetic (e.g. Peano) is complete in the following sense: each formula is in the theory decidable, i.e., ...
Document
... Any valid argument form can be used • there are infinitely many of them, based on different tautologies • validity of an argument form can be verified e.g. using truth tables There are simple, commonly used and useful argument forms • when writing proofs for humans, it is good to use well known argu ...
... Any valid argument form can be used • there are infinitely many of them, based on different tautologies • validity of an argument form can be verified e.g. using truth tables There are simple, commonly used and useful argument forms • when writing proofs for humans, it is good to use well known argu ...
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.