A new proof of Alexeyev`s Theorem
... Hilbert space L2 (X, B, µ) with (X, B, µ) being a standard probability Borel space then there exists a bounded function f ∈ L∞ (X, B, µ) whose spectral measure realizes the maximal spectral type of U . The proof in [1] uses spectral theory and some arguments from the classical theory of analytic fun ...
... Hilbert space L2 (X, B, µ) with (X, B, µ) being a standard probability Borel space then there exists a bounded function f ∈ L∞ (X, B, µ) whose spectral measure realizes the maximal spectral type of U . The proof in [1] uses spectral theory and some arguments from the classical theory of analytic fun ...
Normalised and Cut-free Logic of Proofs
... be obtained in the logic of proofs with an intuitionistic base. Indeed, for Ilp too, we can prove the deduction theorem, the substitution lemma and the internalisation of proofs. Moreover, Ilp is sound and complete with respect to the modal logic S4 with an intuitionistic base, and with respect to H ...
... be obtained in the logic of proofs with an intuitionistic base. Indeed, for Ilp too, we can prove the deduction theorem, the substitution lemma and the internalisation of proofs. Moreover, Ilp is sound and complete with respect to the modal logic S4 with an intuitionistic base, and with respect to H ...
1.4 The set of Real Numbers: Quick Definition and
... We will not discuss those here. p Before we look at the properties of the real numbers, we prove that 2 is irrational using Pythagora’ proof. We begin with a lemma which proof is left as an exercise. Lemma 73 Let n be an integer. If n2 is even then n must also be even. Proof. See exercises. We are n ...
... We will not discuss those here. p Before we look at the properties of the real numbers, we prove that 2 is irrational using Pythagora’ proof. We begin with a lemma which proof is left as an exercise. Lemma 73 Let n be an integer. If n2 is even then n must also be even. Proof. See exercises. We are n ...
Chapter 5.5
... 5-5 Indirect Proof and Inequalities 5-6 Inequalities in Two triangles 5-7 The Pythagorean Theorem 5-8 Applying Special Right Triangles ...
... 5-5 Indirect Proof and Inequalities 5-6 Inequalities in Two triangles 5-7 The Pythagorean Theorem 5-8 Applying Special Right Triangles ...
The Natural Number System: Induction and Counting
... Claim 2. [0, ∞) is inductive. Let x ∈ [0, ∞). Then, by definition of the interval [0, ∞), x ≥ 0. By Axiom 6 of the real number system (and the defining property of 0), x + 1 ≥ 1. We proved earlier that 1 > 0. So by transitivity, x + 1 > 0. Thus, we have shown that [0, ∞) is inductive: if x ∈ [0, ∞), ...
... Claim 2. [0, ∞) is inductive. Let x ∈ [0, ∞). Then, by definition of the interval [0, ∞), x ≥ 0. By Axiom 6 of the real number system (and the defining property of 0), x + 1 ≥ 1. We proved earlier that 1 > 0. So by transitivity, x + 1 > 0. Thus, we have shown that [0, ∞) is inductive: if x ∈ [0, ∞), ...
TEICHIB`S STRONG LAW OF LARGE NUMBERS IN GENERAL
... [I], CBoi- and Sung [Z] and Kuelbs and Zinn [ 5 ] have shown that many classical strong laws of large numbers (SLLN) hold for random variables taking values in a general Banach space under the assumption that the weak law of Iarge numbers (WLLN) holds; this assumption often follows from the geometri ...
... [I], CBoi- and Sung [Z] and Kuelbs and Zinn [ 5 ] have shown that many classical strong laws of large numbers (SLLN) hold for random variables taking values in a general Banach space under the assumption that the weak law of Iarge numbers (WLLN) holds; this assumption often follows from the geometri ...
PPTX
... logic statements by application of equivalence and inference rules, especially in order to massage statements into a desired form. • Devise and attempt multiple different, appropriate strategies for proving a propositional logic statement follows from a list of premises. ...
... logic statements by application of equivalence and inference rules, especially in order to massage statements into a desired form. • Devise and attempt multiple different, appropriate strategies for proving a propositional logic statement follows from a list of premises. ...
Exam 1 Solutions for Spring 2014
... 4. (10 points) A number n is a multiple of 3 if n = 3k for some integer k. Prove that if n2 is a multiple of 3, then n is a multiple of 3. Graded by Stacy Note: This question should have specified that n is an integer. To help compensate for this omission, the lowest score you can receive on this qu ...
... 4. (10 points) A number n is a multiple of 3 if n = 3k for some integer k. Prove that if n2 is a multiple of 3, then n is a multiple of 3. Graded by Stacy Note: This question should have specified that n is an integer. To help compensate for this omission, the lowest score you can receive on this qu ...
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.