Writing Tips
... We will prove this by induction on n. We will prove this by induction on the dimension. We argue by contradiction. Now we consider the converse direction. But f (x) is actually continuous. To see why, consider... The inequality a ≤ b is strict: a < b. Indeed, if there was equality then... ...
... We will prove this by induction on n. We will prove this by induction on the dimension. We argue by contradiction. Now we consider the converse direction. But f (x) is actually continuous. To see why, consider... The inequality a ≤ b is strict: a < b. Indeed, if there was equality then... ...
Introduction to mathematical arguments
... Notice that near the top of the chart, we mention that one can prove a statement by assuming that it is false and deducing a contradiction. This is a useful and fun technique called “proof by contradiction”. Here is how it works. Suppose that we want to prove that the statement P is true. We begin b ...
... Notice that near the top of the chart, we mention that one can prove a statement by assuming that it is false and deducing a contradiction. This is a useful and fun technique called “proof by contradiction”. Here is how it works. Suppose that we want to prove that the statement P is true. We begin b ...
Contents MATH/MTHE 217 Algebraic Structures with Applications Lecture Notes
... There is one intermediary topic that we need to address as we pass from propositional logic to set theory – the topic of quantifiers. There are two quantifiers used in mathematics: • ∃, there exists. • ∀, for all. Much like how propositional logic takes a narrow view of the mathematical world, logic ...
... There is one intermediary topic that we need to address as we pass from propositional logic to set theory – the topic of quantifiers. There are two quantifiers used in mathematics: • ∃, there exists. • ∀, for all. Much like how propositional logic takes a narrow view of the mathematical world, logic ...
Universally true assertions
... If you have proved P(c) for an arbitrary object c of some type, and during the proof have made no restrictions on c, then you are entitled to conclude that P(x) is true for all x of the appropriate type. This process is formalized in mathematical logic as the rule of deduction called universal gener ...
... If you have proved P(c) for an arbitrary object c of some type, and during the proof have made no restrictions on c, then you are entitled to conclude that P(x) is true for all x of the appropriate type. This process is formalized in mathematical logic as the rule of deduction called universal gener ...
full text (.pdf)
... sentences in R A N D O M (a). It is convenient to describe first an alternating PTIME algorithm which decides, for any g, any 0-description D(~), and any formula 0(g) with free variables among g, whether D ~ R 0. We may assume that disjunction and negation are the only connectives and 3 is the only ...
... sentences in R A N D O M (a). It is convenient to describe first an alternating PTIME algorithm which decides, for any g, any 0-description D(~), and any formula 0(g) with free variables among g, whether D ~ R 0. We may assume that disjunction and negation are the only connectives and 3 is the only ...
Generation of Permutations by Addition
... License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ...
... License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ...
Cauchy Sequences
... Exercise 2 Fill the correct definition into box (4). In any metric space S, a divergent Cauchy sequence, because it “converges to a hole,” detects a hole into which S could fit another point. A metric space that has no such holes is called a complete metric space: Definition 4 A metric space S is co ...
... Exercise 2 Fill the correct definition into box (4). In any metric space S, a divergent Cauchy sequence, because it “converges to a hole,” detects a hole into which S could fit another point. A metric space that has no such holes is called a complete metric space: Definition 4 A metric space S is co ...
De Jongh`s characterization of intuitionistic propositional calculus
... The problem of a syntactic characterization of intuitionistic propositional calculus IPC goes back to Lukasiewicz. In [14], Lukasiewicz conjectured that IPC is the only intermediate logic1 having the disjunction property, i.e., if ` φ ∨ ψ, then ` φ or ` ψ. However, Kreisel and Putnam [13] disproved ...
... The problem of a syntactic characterization of intuitionistic propositional calculus IPC goes back to Lukasiewicz. In [14], Lukasiewicz conjectured that IPC is the only intermediate logic1 having the disjunction property, i.e., if ` φ ∨ ψ, then ` φ or ` ψ. However, Kreisel and Putnam [13] disproved ...
On an Integer Sequence Related to a Product Combinatorial Relevance
... Note that the property that the coefficient of xi in Gn (x) is the number of distinct subsets of {1, . . . , n} whose elements sum to i was used by Friedman and Keith [5] to deduce a necessary and sufficient condition for the existence of a basic (n,k) magic carpet. Stanley [9], using the “hard Lefs ...
... Note that the property that the coefficient of xi in Gn (x) is the number of distinct subsets of {1, . . . , n} whose elements sum to i was used by Friedman and Keith [5] to deduce a necessary and sufficient condition for the existence of a basic (n,k) magic carpet. Stanley [9], using the “hard Lefs ...
CONGRUENCES Modular arithmetic. Two whole numbers a and b
... Problem 6. Carry out the Euclidean algorithm for the pair a = 245, b = 193. Theorem 7. (Bezout) If d = gcd(a, b), then there are exist integers k and l such that d = ka + lb. In particular, for any pair of numbers a and b which are relatively prime, one can write 1 = ka + lb for some integers k and ...
... Problem 6. Carry out the Euclidean algorithm for the pair a = 245, b = 193. Theorem 7. (Bezout) If d = gcd(a, b), then there are exist integers k and l such that d = ka + lb. In particular, for any pair of numbers a and b which are relatively prime, one can write 1 = ka + lb for some integers k and ...
Aristotle`s particularisation
... of a formal Arithmetical language interprets as an arithmetical relation that is denoted in the interpretation by ‘P ∗ (x)’, and the formula ‘[¬(∀x)¬P (x)]’ as the arithmetical proposition denoted by ‘¬(∀x)¬P ∗ (x)’, the formula ‘[(∃x)P (x)]’ cannot be assumed to always interpret as the arithmetical ...
... of a formal Arithmetical language interprets as an arithmetical relation that is denoted in the interpretation by ‘P ∗ (x)’, and the formula ‘[¬(∀x)¬P (x)]’ as the arithmetical proposition denoted by ‘¬(∀x)¬P ∗ (x)’, the formula ‘[(∃x)P (x)]’ cannot be assumed to always interpret as the arithmetical ...
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.