Chapter 8 Number Theory 8-1 Prime Numbers and Composite N
... Eg. Let S Z, and S has 37 elements. Then S contains two elements that have the same remainder upon division by 36. (Proof) n=36q+r, 0≦r<36. There are 36 possible values of r. According to the pigeonhole principle, the result is established. Eg. Any subset of size six from S={1,2,3,4,5,6,7,8,9} mus ...
... Eg. Let S Z, and S has 37 elements. Then S contains two elements that have the same remainder upon division by 36. (Proof) n=36q+r, 0≦r<36. There are 36 possible values of r. According to the pigeonhole principle, the result is established. Eg. Any subset of size six from S={1,2,3,4,5,6,7,8,9} mus ...
Logic, Proofs, and Sets
... P if and only if Q has the same meaning as the statement if P, then Q and if Q, then P. This statement asserts a kind of equality – that P and Q have the same meaning: P is true exactly when Q is. The phrase if and only if is frequently abbreviated iff, especially in definitions. The mathematical sy ...
... P if and only if Q has the same meaning as the statement if P, then Q and if Q, then P. This statement asserts a kind of equality – that P and Q have the same meaning: P is true exactly when Q is. The phrase if and only if is frequently abbreviated iff, especially in definitions. The mathematical sy ...
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... In [2], Church proves that the following formulation of one-half of Leibniz’s characterization of equality holds in pure predicate calculus F1: Definition 1. Substitution of equals for equals: If S results from R by substitution of Q for P at one or more places in R (not necessarily at all occurrenc ...
... In [2], Church proves that the following formulation of one-half of Leibniz’s characterization of equality holds in pure predicate calculus F1: Definition 1. Substitution of equals for equals: If S results from R by substitution of Q for P at one or more places in R (not necessarily at all occurrenc ...
A SHORT AND READABLE PROOF OF CUT ELIMINATION FOR
... step we prove the case for m > 0. Cases (2)–(8) are numbered by the rule number (Definition 2.1) of the last rule applied in deriving (Γ ⊢ ∆)[a]. (2) (Γ ⊢ ∆)[a] = Γ[a], A[a] → B[a] ⊢ ∆[a]. Thus the premises of the rule,8 Γ[a], A[a] → ⊥ ⊢ ∆[a] and Γ[a], B[a] ⊢ ∆[a], are each derived with orders < m. ...
... step we prove the case for m > 0. Cases (2)–(8) are numbered by the rule number (Definition 2.1) of the last rule applied in deriving (Γ ⊢ ∆)[a]. (2) (Γ ⊢ ∆)[a] = Γ[a], A[a] → B[a] ⊢ ∆[a]. Thus the premises of the rule,8 Γ[a], A[a] → ⊥ ⊢ ∆[a] and Γ[a], B[a] ⊢ ∆[a], are each derived with orders < m. ...
the Catalan numbers
... =O n j=1 n2 In case the λ(n) ’s were n i.i.d. random variables, one would rather expect this variance to be O(1/n). But the eigenvalues of a random matrix are far from being i.i.d. in general, as already observed when computing their joint distribution in Lecture 6 in the Gaussian case. They are act ...
... =O n j=1 n2 In case the λ(n) ’s were n i.i.d. random variables, one would rather expect this variance to be O(1/n). But the eigenvalues of a random matrix are far from being i.i.d. in general, as already observed when computing their joint distribution in Lecture 6 in the Gaussian case. They are act ...
Class Notes
... might be thought of as an abstract object existing only in our minds, the fact is that mathematics advances only in so far as proofs are communicated. And writing remains the principal means of such communication. So to be a mathematician, you need to learn how to prove things but also to write thos ...
... might be thought of as an abstract object existing only in our minds, the fact is that mathematics advances only in so far as proofs are communicated. And writing remains the principal means of such communication. So to be a mathematician, you need to learn how to prove things but also to write thos ...
Exploring Fibonacci Numbers
... The Fibonacci numbers have a myriad of interesting algebraic properties. Many such properties can be proven in more than one way, as with the Binet form alone. As demonstrated in this section, the recursive definition of the Fibonacci numbers lends itself well to proofs by induction. Property 6.1. F ...
... The Fibonacci numbers have a myriad of interesting algebraic properties. Many such properties can be proven in more than one way, as with the Binet form alone. As demonstrated in this section, the recursive definition of the Fibonacci numbers lends itself well to proofs by induction. Property 6.1. F ...
Constructive Set Theory and Brouwerian Principles1
... the proof consists of defining a transfinite type structure over a special combinatory algebra whose domain is the set of all arithmetical functions from N to N with application being continuous function application in the sense of Kleene’s second algebra K2 . The transfinite type structure serves t ...
... the proof consists of defining a transfinite type structure over a special combinatory algebra whose domain is the set of all arithmetical functions from N to N with application being continuous function application in the sense of Kleene’s second algebra K2 . The transfinite type structure serves t ...
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.