ppt - Purdue College of Engineering
... Deduction Method in Proofs • When proving P Q… – add P to premises and prove Q. ...
... Deduction Method in Proofs • When proving P Q… – add P to premises and prove Q. ...
math8_breakout_aug_24_v1
... 8.G.B Understand and apply the Pythagorean Theorem. 8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse. 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.G.B.8 Apply the P ...
... 8.G.B Understand and apply the Pythagorean Theorem. 8.G.B.6 Explain a proof of the Pythagorean Theorem and its converse. 8.G.B.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. 8.G.B.8 Apply the P ...
Objective - To use properties of numbers in proofs.
... Conjecture - a statement or conditional that one is trying to prove. Types of supportive statements used in proofs 1) Undefined terms - Terminology so fundamental it defies definition. ie: point, line, straight, etc. 2) Definitions - Statements defined by other terms. ie: A quadrilateral is a 4 side ...
... Conjecture - a statement or conditional that one is trying to prove. Types of supportive statements used in proofs 1) Undefined terms - Terminology so fundamental it defies definition. ie: point, line, straight, etc. 2) Definitions - Statements defined by other terms. ie: A quadrilateral is a 4 side ...
Maximizing the number of nonnegative subsets, SIAM J. Discrete
... Acknowledgments. We thank Emmanuel Tsukerman for telling us about the problem considered here, and Benny Sudakov for fruitful discussions, useful suggestions, and helpful ideas. Note added in proof. Alexey Pokrovskiy recently informed us that he has found an alternative proof of Theorem 3.1. His pro ...
... Acknowledgments. We thank Emmanuel Tsukerman for telling us about the problem considered here, and Benny Sudakov for fruitful discussions, useful suggestions, and helpful ideas. Note added in proof. Alexey Pokrovskiy recently informed us that he has found an alternative proof of Theorem 3.1. His pro ...
geometry, probability, and cardinality
... When we fail, we will see that A has more elements. Specifically, we associate 1 ∈ A to 1 ∈ B, we associate 2 ∈ A to 2 ∈ B, and so on, until we finally associate m ∈ A to m ∈ B. At this point, each element of B has been associated to a distinct element of A. If we try to continue the process, we see ...
... When we fail, we will see that A has more elements. Specifically, we associate 1 ∈ A to 1 ∈ B, we associate 2 ∈ A to 2 ∈ B, and so on, until we finally associate m ∈ A to m ∈ B. At this point, each element of B has been associated to a distinct element of A. If we try to continue the process, we see ...
Full text - The Fibonacci Quarterly
... for some « E { 1 , . . . , k}. E x a m p l e . Consider 9{x) = ( 3 * i ) . . . (3Yn)[(X + 1 .= Y?) A • • • A (X + n = Yn2)] so that 9(X) asserts that X - f l , X + 2 , . . . , X + n are perfect squares within the field. In this case one can take k = 2 with /xi = l / 2 n and / j 2 = 1. The first valu ...
... for some « E { 1 , . . . , k}. E x a m p l e . Consider 9{x) = ( 3 * i ) . . . (3Yn)[(X + 1 .= Y?) A • • • A (X + n = Yn2)] so that 9(X) asserts that X - f l , X + 2 , . . . , X + n are perfect squares within the field. In this case one can take k = 2 with /xi = l / 2 n and / j 2 = 1. The first valu ...
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION
... that xn → x when n → ∞; then f (xn ) = 0 for every n and the sequence {f (xn )} does not converge to f (x) = 1/q, so f is not continuous at x. On the other hand, for x ∈ R \ Q, let us see that f is continuous at x by checking that f (xn ) → f (x) = 0 for every sequence {xn } that tends to x. As f (y ...
... that xn → x when n → ∞; then f (xn ) = 0 for every n and the sequence {f (xn )} does not converge to f (x) = 1/q, so f is not continuous at x. On the other hand, for x ∈ R \ Q, let us see that f is continuous at x by checking that f (xn ) → f (x) = 0 for every sequence {xn } that tends to x. As f (y ...
The Ramanujan-Nagell Theorem: Understanding the Proof 1
... that were considered belonged to an imaginary quadratic field. In this manner, many questions seemingly posed in number theory have proofs in ANT, such as Wile’s famous proof of Fermat’s Last Theorem in 1994. Ideas extending previous concepts are extremely useful to the progress of mathematics, and ...
... that were considered belonged to an imaginary quadratic field. In this manner, many questions seemingly posed in number theory have proofs in ANT, such as Wile’s famous proof of Fermat’s Last Theorem in 1994. Ideas extending previous concepts are extremely useful to the progress of mathematics, and ...
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.