Advanced Topics in Propositional Logic
... Go through these, and whenever you encounter Ai such that neither it nor its negation is derivable from , add Ai to . In view of Lemma 5, you will end up with a formally complete set. To see that the same set is also formally consistent, suppose, for a contradiction, that it is not. Consider the s ...
... Go through these, and whenever you encounter Ai such that neither it nor its negation is derivable from , add Ai to . In view of Lemma 5, you will end up with a formally complete set. To see that the same set is also formally consistent, suppose, for a contradiction, that it is not. Consider the s ...
1 Counting mappings
... r with the oldest son s. We assume that T1 contains r, and T2 contains s. It is not hard to check that this is a 1-1 correspondence. Hence, for the number tn of planar rooted trees with n edges, we have t0 = 1 and for every n > 0, tn := t0 tn−1 + t1 tn−2 + . . . + tn−1 t0 . In other words, tn = cn f ...
... r with the oldest son s. We assume that T1 contains r, and T2 contains s. It is not hard to check that this is a 1-1 correspondence. Hence, for the number tn of planar rooted trees with n edges, we have t0 = 1 and for every n > 0, tn := t0 tn−1 + t1 tn−2 + . . . + tn−1 t0 . In other words, tn = cn f ...
Vol.16 No.1 - Department of Mathematics
... Otherwise, there exists a segment AmBm intersecting M at more than 1 point. Let it intersect the perimeter of M again at Dm. Since AiBi’s do not intersect, so AjDj’s (being subsets of AiBi’s) do not intersect. In particular, Dm is not a vertex of M. Now AmDm divides the perimeter of M into two parts ...
... Otherwise, there exists a segment AmBm intersecting M at more than 1 point. Let it intersect the perimeter of M again at Dm. Since AiBi’s do not intersect, so AjDj’s (being subsets of AiBi’s) do not intersect. In particular, Dm is not a vertex of M. Now AmDm divides the perimeter of M into two parts ...
The Repeated Sums of Integers
... distribution map from a source S to a destination T, with capacities indicated by the weights on the edges. Since the water is not escaping the system, the total output at S must equal the total input at T, i.e., ...
... distribution map from a source S to a destination T, with capacities indicated by the weights on the edges. Since the water is not escaping the system, the total output at S must equal the total input at T, i.e., ...
5.2 The Master Theorem
... Removing Floors and Ceilings We have also pointed out that a more realistic Master Theorem would apply to recurrences of the form T (n) = aT (n/b) + nc , or T (n) = aT (n/b) + nc , or even T (n) = a T (n/b) + (a − a )T (n/b) + nc . For example, if we are applying mergesort to an array of s ...
... Removing Floors and Ceilings We have also pointed out that a more realistic Master Theorem would apply to recurrences of the form T (n) = aT (n/b) + nc , or T (n) = aT (n/b) + nc , or even T (n) = a T (n/b) + (a − a )T (n/b) + nc . For example, if we are applying mergesort to an array of s ...
Logic and Mathematical Reasoning
... In this chapter we introduce classical logic which has two truth values, True and False. Every proposition takes on a single truth value. Definition 1.1.1 (Proposition). A proposition is a sentence that is either true or false. Definition 1.1.2 (Conjunction, Disjunction, Negation). Given proposition ...
... In this chapter we introduce classical logic which has two truth values, True and False. Every proposition takes on a single truth value. Definition 1.1.1 (Proposition). A proposition is a sentence that is either true or false. Definition 1.1.2 (Conjunction, Disjunction, Negation). Given proposition ...
Lecture notes, sections 2.1 to 2.3
... The set ha, bi is the set of all multiples of the generator d. Since a and b are elements of ha, bi, both a and b are multiples of d. Therefore, (by the second fact above), d must divide GCD(a, b). Now d and GCD(a, b) are positive integers dividing each other, so they must be equal (third fact above ...
... The set ha, bi is the set of all multiples of the generator d. Since a and b are elements of ha, bi, both a and b are multiples of d. Therefore, (by the second fact above), d must divide GCD(a, b). Now d and GCD(a, b) are positive integers dividing each other, so they must be equal (third fact above ...
Chapter 1 Introduction to prime number theory
... with a not so difficult proof based on complex analysis alone and avoiding functional analysis. In this course, we prove the Tauberian theorem via Newman’s method, and deduce from this the Prime Number Theorem as well as the Prime Number Theorem for arithmetic progressions (see below). ...
... with a not so difficult proof based on complex analysis alone and avoiding functional analysis. In this course, we prove the Tauberian theorem via Newman’s method, and deduce from this the Prime Number Theorem as well as the Prime Number Theorem for arithmetic progressions (see below). ...
Sample pages 1 PDF
... 2. Groupoids, semigroups, and groups. Algebras A = (A, ◦) with an operation ◦ : A2 → A are termed groupoids. If ◦ is associative then A is called a semigroup, and if ◦ is additionally invertible, then A is said to be a group. It is provable that a group (G, ◦) in this sense contains exactly one unit ...
... 2. Groupoids, semigroups, and groups. Algebras A = (A, ◦) with an operation ◦ : A2 → A are termed groupoids. If ◦ is associative then A is called a semigroup, and if ◦ is additionally invertible, then A is said to be a group. It is provable that a group (G, ◦) in this sense contains exactly one unit ...
PRIMES OF THE FORM x2 + ny 2 AND THE GEOMETRY OF
... 2. Minkowski’s Geometry of Numbers and Convenient Numbers An amazing discovery of 19th -century mathematics was Minkowski’s simple use of geometry to prove theorems in about numbers. We will apply his methods to reprove Gauss’s result that the 65 numbers in Table 1 found by Euler are all convenient ...
... 2. Minkowski’s Geometry of Numbers and Convenient Numbers An amazing discovery of 19th -century mathematics was Minkowski’s simple use of geometry to prove theorems in about numbers. We will apply his methods to reprove Gauss’s result that the 65 numbers in Table 1 found by Euler are all convenient ...
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.