Lecture 39 Notes
... that A ⇒ A is uniformly true. T P (x) is ∀[x : D].P (x), and we know that the evidence is uniform in x as in x:D ∀[x : D].(P (x) ⇒ P (x)) or ∀[x : D].(P (x) & Q(x) ⇒ P (x)). In Lecture 38 there is a discussion of the close connection between programs with assertions (asserted programs) justified by ...
... that A ⇒ A is uniformly true. T P (x) is ∀[x : D].P (x), and we know that the evidence is uniform in x as in x:D ∀[x : D].(P (x) ⇒ P (x)) or ∀[x : D].(P (x) & Q(x) ⇒ P (x)). In Lecture 38 there is a discussion of the close connection between programs with assertions (asserted programs) justified by ...
lecture1.5
... If we are asked to show that a proposition is False, then we just need to provide one counter-example for which the proposition is False In other words, to show that x P(x) is False, we can just show x P(x) = x P(x) to be True Example: “Every positive integer is the sum of the squares of two in ...
... If we are asked to show that a proposition is False, then we just need to provide one counter-example for which the proposition is False In other words, to show that x P(x) is False, we can just show x P(x) = x P(x) to be True Example: “Every positive integer is the sum of the squares of two in ...
Assignment 6
... (2) If we apply the minimization operator to a function f (x, y) that is always positive at x, e.g. ∀y. f (x, y) 6= 0, then it does not produce a value but “diverges,” on some input x. The domain of such a function µy.f (x, y) = 0 is {x : N | ∃y. f (x, y) = 0}. Note, we can represent λx.µy.f (x, y) ...
... (2) If we apply the minimization operator to a function f (x, y) that is always positive at x, e.g. ∀y. f (x, y) 6= 0, then it does not produce a value but “diverges,” on some input x. The domain of such a function µy.f (x, y) = 0 is {x : N | ∃y. f (x, y) = 0}. Note, we can represent λx.µy.f (x, y) ...
A Primer on Mathematical Proof
... Common Mistake # 2. Checking the conclusion in a few specific examples, and extrapolating that it always holds. A proof must give a rigorous argument that the conclusion holds in all situations satisfying the hypotheses. For example, if you want to prove that some property holds for all real numbers ...
... Common Mistake # 2. Checking the conclusion in a few specific examples, and extrapolating that it always holds. A proof must give a rigorous argument that the conclusion holds in all situations satisfying the hypotheses. For example, if you want to prove that some property holds for all real numbers ...
Mathematical Induction - Singapore Mathematical Society
... called Pascal's Triangle. The first is a Chinese version copied from a diagram that appeared in the Ssu-yii.an yii.-chien (Precious Mirror of the Four Elements) by Chu Shih-chieh in 1303. Chu disclaims credit for the triangle and it seems likely that it originated in China about 1100. Note the use o ...
... called Pascal's Triangle. The first is a Chinese version copied from a diagram that appeared in the Ssu-yii.an yii.-chien (Precious Mirror of the Four Elements) by Chu Shih-chieh in 1303. Chu disclaims credit for the triangle and it seems likely that it originated in China about 1100. Note the use o ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.