2-6 Algebraic Proof
... plane R, and B is on . Which option states the postulate that can be used to show that A, H, and D are coplanar? A. Through any two points on the same line, there is exactly one plane. B. Through any three points not on the same line, there is exactly one plane. C. If two points lie in a plane, then ...
... plane R, and B is on . Which option states the postulate that can be used to show that A, H, and D are coplanar? A. Through any two points on the same line, there is exactly one plane. B. Through any three points not on the same line, there is exactly one plane. C. If two points lie in a plane, then ...
Predicate Calculus - National Taiwan University
... possible interpretations (all the possible functions and predicates). But if the domain is infinite? Intuitively, this is why a computer cannot be programmed to determine if an arbitrary formula in predicate logic is a tautology (for ...
... possible interpretations (all the possible functions and predicates). But if the domain is infinite? Intuitively, this is why a computer cannot be programmed to determine if an arbitrary formula in predicate logic is a tautology (for ...
Remainder Theorem and Factor Theorem
... If r is a real zero of a polynomial function f, then • f(r) = 0 • r is an x-intercept of the graph of f • r is a solution of the equation f(x) = 0 • r is also called a root of the equation • (x – r) is a factor of the polynomial • if you divide the polynomial by (x –r), the remainder would be zero. ...
... If r is a real zero of a polynomial function f, then • f(r) = 0 • r is an x-intercept of the graph of f • r is a solution of the equation f(x) = 0 • r is also called a root of the equation • (x – r) is a factor of the polynomial • if you divide the polynomial by (x –r), the remainder would be zero. ...
The Asymptotic Density of Relatively Prime Pairs and of Square
... Let Cn D fk W 1 k n; k is square-freeg. If limn!1 jCnn j exists, we call this limit the asymptotic density of square-free numbers. After giving the number theoretic proof of Theorem 2.1, we will prove the following theorem. Theorem 2.2. The asymptotic density of square-free integers is ...
... Let Cn D fk W 1 k n; k is square-freeg. If limn!1 jCnn j exists, we call this limit the asymptotic density of square-free numbers. After giving the number theoretic proof of Theorem 2.1, we will prove the following theorem. Theorem 2.2. The asymptotic density of square-free integers is ...
patterns in continued fraction expansions
... expansions in different bases the only thing that changes is how we represent those integers. Whether or not the expansion is finite or infinite does not change, even if we do change the base. For example, in base 10, 31/25 has continued fraction expansion [1,4,6], the expansion of 1/3 is [0,3], and ...
... expansions in different bases the only thing that changes is how we represent those integers. Whether or not the expansion is finite or infinite does not change, even if we do change the base. For example, in base 10, 31/25 has continued fraction expansion [1,4,6], the expansion of 1/3 is [0,3], and ...
Algebraic Laws for Nondeterminism and Concurrency
... that it is natural; to this end, we give an alternative characterization of the resulting equivalence relation in Section 2.2. In the caseof deterministic sequential programs, the behavior of a program p is usually taken to be its input-output function IO(p). Here, an observation of p is taken to be ...
... that it is natural; to this end, we give an alternative characterization of the resulting equivalence relation in Section 2.2. In the caseof deterministic sequential programs, the behavior of a program p is usually taken to be its input-output function IO(p). Here, an observation of p is taken to be ...
Math 13 — An Introduction to Abstract Mathematics October 20, 2014
... distribution of prime numbers were studied for over 2000 years before, arguably, any serious applications were discovered. Sometimes a real-world problem motivates generalizations that have no obvious application, and may never do so. For example, the geometry of projecting 3D objects onto a 2D scre ...
... distribution of prime numbers were studied for over 2000 years before, arguably, any serious applications were discovered. Sometimes a real-world problem motivates generalizations that have no obvious application, and may never do so. For example, the geometry of projecting 3D objects onto a 2D scre ...
Math `Convincing and Proving` Critiquing
... Attempt 1, although a logical approach, assumes what we are setting out to prove. If the argument were reversed, it would provide the basis for a proof. Attempt 2 is correct. Attempt 3 is the basis for a very elegant proof, but there are some holes and unnecessary jumps in it at present. The first i ...
... Attempt 1, although a logical approach, assumes what we are setting out to prove. If the argument were reversed, it would provide the basis for a proof. Attempt 2 is correct. Attempt 3 is the basis for a very elegant proof, but there are some holes and unnecessary jumps in it at present. The first i ...
Inference and Proofs - Dartmouth Math Home
... We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients. First, we introduced symbols for members of the universe of integers. In other words, rather than saying “suppose we have two integers,” we introduced symbols for th ...
... We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients. First, we introduced symbols for members of the universe of integers. In other words, rather than saying “suppose we have two integers,” we introduced symbols for th ...
Factoring Out the Impossibility of Logical Aggregation
... same individuals in two pro…les, it is a member of either both or none of the two social sets obtained from the mapping. More informally, one can decide whether a formula belongs to a social set just by considering which individual sets it belongs to, regardless of the other formulas that these sets ...
... same individuals in two pro…les, it is a member of either both or none of the two social sets obtained from the mapping. More informally, one can decide whether a formula belongs to a social set just by considering which individual sets it belongs to, regardless of the other formulas that these sets ...
Birkhoff`s variety theorem in many sorts
... of finitary logic and start working in the logic Lω,λ allowing quantifications of less than λ variables where λ > card S. However, the “naive” generalization without directed unions appears rather persistently in books and papers. (See e.g. page 141 of [1], page 105 of [4], page 107 of [5] and page 24 ...
... of finitary logic and start working in the logic Lω,λ allowing quantifications of less than λ variables where λ > card S. However, the “naive” generalization without directed unions appears rather persistently in books and papers. (See e.g. page 141 of [1], page 105 of [4], page 107 of [5] and page 24 ...
On the prime factors of the number 2 p-1 - 1
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Theorem
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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.