Inclusion-Exclusion Principle and Applications
... Answer: Let S = {1, 2, 3, . . . , 2009}. A1 = {4, 8, 12, . . . , 2008}, |A1 | = b 2009 4 c = 502. A2 = {5, 10, 15, . . . , 2005}, |A2 | = b 2009 5 c = 401. A3 = {6, 12, 18, . . . , 2004}, |A3 | = b 2009 6 c = 334. The least common multiple of some numbers a1 , a2 , . . . , an is the smallest positiv ...
... Answer: Let S = {1, 2, 3, . . . , 2009}. A1 = {4, 8, 12, . . . , 2008}, |A1 | = b 2009 4 c = 502. A2 = {5, 10, 15, . . . , 2005}, |A2 | = b 2009 5 c = 401. A3 = {6, 12, 18, . . . , 2004}, |A3 | = b 2009 6 c = 334. The least common multiple of some numbers a1 , a2 , . . . , an is the smallest positiv ...
Language, Proof and Logic
... domain, we can use universal generalization to get ∀x [P(x) → Q(x)]. This is how formal systems of deduction can get by without having an explicit rule of general conditional proof. One could in a sense think of universal generalization as a special case of general conditional proof. After all, if w ...
... domain, we can use universal generalization to get ∀x [P(x) → Q(x)]. This is how formal systems of deduction can get by without having an explicit rule of general conditional proof. One could in a sense think of universal generalization as a special case of general conditional proof. After all, if w ...
Partial Correctness Specification
... A proof in Floyd-Hoare logic is a sequence of lines, each of which is either an axiom of the logic or follows from earlier lines by a rule of inference of the logic u ...
... A proof in Floyd-Hoare logic is a sequence of lines, each of which is either an axiom of the logic or follows from earlier lines by a rule of inference of the logic u ...
Methods and Formal Models / Nachum Dershowitz
... the concurrent occurrences of P, P1||P2||…||Pn, so that it’s not used by more times than its limit. Example Suppose we have a program P, and we would like to run it in several parallel occurrences. Suppose also, that P uses some kind of a common resource, which is limited in the system. Let P be the ...
... the concurrent occurrences of P, P1||P2||…||Pn, so that it’s not used by more times than its limit. Example Suppose we have a program P, and we would like to run it in several parallel occurrences. Suppose also, that P uses some kind of a common resource, which is limited in the system. Let P be the ...
CS 40: Foundations of Computer Science
... take d to be true as well, then both of our assumptions are true. There fore this conclusion is not valid. c)The issue is ¬ e V d, which is equivalent to the conditional statement e → d. This does not follow from our assumptions. If we take d to be false, e to be true, and s to be false, then this p ...
... take d to be true as well, then both of our assumptions are true. There fore this conclusion is not valid. c)The issue is ¬ e V d, which is equivalent to the conditional statement e → d. This does not follow from our assumptions. If we take d to be false, e to be true, and s to be false, then this p ...
A Logic of Belief with the Complexity Measure
... B r = {ir , sr }, where ir is an initial belief set – beliefs that are initially actively hold by the agent, and sr = {α | c(α | ir ) ≤ r} is a potential belief set – beliefs for which an agent has a resource to infer them from his initial beliefs. In the logic of belief with complexity (lbc), the k ...
... B r = {ir , sr }, where ir is an initial belief set – beliefs that are initially actively hold by the agent, and sr = {α | c(α | ir ) ≤ r} is a potential belief set – beliefs for which an agent has a resource to infer them from his initial beliefs. In the logic of belief with complexity (lbc), the k ...
NUMBER THEORY 1. Divisor Counting Theorem 1. A number is a
... Proof. If a number is divisible by 6, it is composite; thus no prime number is 0 mod 6. If a number is 2 more than a multiple of 6, it is even, because all multiples of 6 are even. The only even prime is 2. Thus a prime number cannot be 2 mod 6 unless it is actually equal to 2. Similarly, if a numbe ...
... Proof. If a number is divisible by 6, it is composite; thus no prime number is 0 mod 6. If a number is 2 more than a multiple of 6, it is even, because all multiples of 6 are even. The only even prime is 2. Thus a prime number cannot be 2 mod 6 unless it is actually equal to 2. Similarly, if a numbe ...
Quadrature of the parabola with the square pyramidal number
... Split the segments AB and BC into six equal parts and lead, from split points on AB parallels to BC, and from points on BC lines joining with A. The parabola passes through the points of intersection of the grid, as drawn, because, for one of its properties, it cuts the vertical lines of the grid in ...
... Split the segments AB and BC into six equal parts and lead, from split points on AB parallels to BC, and from points on BC lines joining with A. The parabola passes through the points of intersection of the grid, as drawn, because, for one of its properties, it cuts the vertical lines of the grid in ...
The Dirichlet Unit Theorem
... the positive units of B0 form a B0 = Z[ m] of the ring B of algebraic integers, and √ subgroup H of the positive units of B. Let u = 12 (a + b m) be the fundamental unit of the number field L. 2. If a and b are both even, for example when m = 17, show that H consists of the powers of u, in other word ...
... the positive units of B0 form a B0 = Z[ m] of the ring B of algebraic integers, and √ subgroup H of the positive units of B. Let u = 12 (a + b m) be the fundamental unit of the number field L. 2. If a and b are both even, for example when m = 17, show that H consists of the powers of u, in other word ...
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.