Prolog 1 - Department of Computer Science
... • The “fool” says in his heart “There is no God” (see Psalm 14). But is he/she talking about “that than which nothing greater can be thought”? Because, if the “fool” can conceive of such a being, this being exists in his or her understanding, therefore . . . • A monk named Gaunilo, complained that i ...
... • The “fool” says in his heart “There is no God” (see Psalm 14). But is he/she talking about “that than which nothing greater can be thought”? Because, if the “fool” can conceive of such a being, this being exists in his or her understanding, therefore . . . • A monk named Gaunilo, complained that i ...
MATH 311W Wksht 1 • A logical statement is a phrase that is
... NOTE: In logic, we care a lot about when two statements are equivalent, the statements themselves do not always have to be true. When we prove theorems, we will care about statements being true. HW # 1: [Read N&B: pg 2 (definition of statement), and pg 25-27 look at Ex. 1-10, pg 30] 1. Explain wheth ...
... NOTE: In logic, we care a lot about when two statements are equivalent, the statements themselves do not always have to be true. When we prove theorems, we will care about statements being true. HW # 1: [Read N&B: pg 2 (definition of statement), and pg 25-27 look at Ex. 1-10, pg 30] 1. Explain wheth ...
Why a Negative Number Times a Negative Number Equals a
... An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, this number system never ends or repeats. The Pythagorean discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers. It turns out that the rationa ...
... An irrational number is a number that cannot be written as a ratio (or fraction). In decimal form, this number system never ends or repeats. The Pythagorean discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers. It turns out that the rationa ...
Soundness and Completeness - Cognitive Science Department
... By inductive assumption: h(i) = True iff EG i So, h(1 2) = True iff h(1) = True and h(1) = True iff EG 1 and EG 2 iff (trivial) EG 1 2 iff EG Case 2: = ’ By ind. assumption: h(’) = true iff EG ’ Hence, h() = True iff h(’) = false iff not EG ’ iff (Either-Or L ...
... By inductive assumption: h(i) = True iff EG i So, h(1 2) = True iff h(1) = True and h(1) = True iff EG 1 and EG 2 iff (trivial) EG 1 2 iff EG Case 2: = ’ By ind. assumption: h(’) = true iff EG ’ Hence, h() = True iff h(’) = false iff not EG ’ iff (Either-Or L ...
Logic and Discrete Mathematics for Computer Scientists
... The formal approach to the presentation of material has, we believe, a number of significant advantages, especially for Computer Science students, but also, for more traditional math students who might find their way into the course. In mathematics departments proofs are typically learned by student ...
... The formal approach to the presentation of material has, we believe, a number of significant advantages, especially for Computer Science students, but also, for more traditional math students who might find their way into the course. In mathematics departments proofs are typically learned by student ...
A Simple Proof of the Aztec Diamond Theorem
... Proof: Given a tiling T of ADn , we associate T with an n-tuple (τ1 , . . . , τn ) of nonintersecting paths as follows. Let the rows of ADn be indexed by 1, 2, . . . , 2n from bottom to top. For each i, (1 ≤ i ≤ n) we define a path τi from the center of the left-hand edge of the ith row to the cente ...
... Proof: Given a tiling T of ADn , we associate T with an n-tuple (τ1 , . . . , τn ) of nonintersecting paths as follows. Let the rows of ADn be indexed by 1, 2, . . . , 2n from bottom to top. For each i, (1 ≤ i ≤ n) we define a path τi from the center of the left-hand edge of the ith row to the cente ...
On Action Logic
... for *. The cut-elimination theorem and the decidability of this system are left open. We show that the cut-elimination theorem does not hold for Jipsen’s system. We consider logic ACTω, i.e. the equational theory ∗−continuous action algebras (roughly: action algebras with a standard ∗−operation). W ...
... for *. The cut-elimination theorem and the decidability of this system are left open. We show that the cut-elimination theorem does not hold for Jipsen’s system. We consider logic ACTω, i.e. the equational theory ∗−continuous action algebras (roughly: action algebras with a standard ∗−operation). W ...
4.1 BASICS OF COUNTING
... deck? Answer. 52 · 51 · 50 · 49 · 48 = 525 review §1.8: falling power xr = x(x − 1)(x − 2) · · · (x − r + 1). Remark: In general, falling powers count ordered selections. §4.4 shows how binomial coefficients count subsets (i.e., unordered selections). ...
... deck? Answer. 52 · 51 · 50 · 49 · 48 = 525 review §1.8: falling power xr = x(x − 1)(x − 2) · · · (x − r + 1). Remark: In general, falling powers count ordered selections. §4.4 shows how binomial coefficients count subsets (i.e., unordered selections). ...
- ScholarWorks@GVSU
... the appropriate mathematical background). A proof must use correct, logical reasoning and be based on previously established results. These previous results can be axioms, definitions, or previously proven theorems. These terms are discussed below. Surprising to some is the fact that in mathematics, ...
... the appropriate mathematical background). A proof must use correct, logical reasoning and be based on previously established results. These previous results can be axioms, definitions, or previously proven theorems. These terms are discussed below. Surprising to some is the fact that in mathematics, ...
(pdf)
... of Euler's Pentagonal Number Theorem. A partition λ= (λ1 λ2 . . . λr ) of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. As each n is nite, it follows trivially that each sequence is nite. Elements of a partition are called parts. As it is common for a given ...
... of Euler's Pentagonal Number Theorem. A partition λ= (λ1 λ2 . . . λr ) of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. As each n is nite, it follows trivially that each sequence is nite. Elements of a partition are called parts. As it is common for a given ...
- ScholarWorks@GVSU
... the appropriate mathematical background). A proof must use correct, logical reasoning and be based on previously established results. These previous results can be axioms, definitions, or previously proven theorems. These terms are discussed below. Surprising to some is the fact that in mathematics, ...
... the appropriate mathematical background). A proof must use correct, logical reasoning and be based on previously established results. These previous results can be axioms, definitions, or previously proven theorems. These terms are discussed below. Surprising to some is the fact that in mathematics, ...
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.