Chapter 2 Propositional Logic
... By definition p ↔ q means (p → q) ∧ (q → p). This statement, also referred to as if and only if, appears very often during proofs. Finally, when combining logic operators, there may be doubts on which operator to apply first. The use of parentheses helps in deciding which operators should be compute ...
... By definition p ↔ q means (p → q) ∧ (q → p). This statement, also referred to as if and only if, appears very often during proofs. Finally, when combining logic operators, there may be doubts on which operator to apply first. The use of parentheses helps in deciding which operators should be compute ...
HKT Chapters 1 3
... A strict partial order is a binary relation < that is irreflexive and transitive. Any strict partial order < has an associated partial order ≤ defined by a ≤ b if a < b or a = b. Any preorder ≤ has an associated strict partial order defined by a < b if a ≤ b but b ≤ a. For partial orders ≤, these two ...
... A strict partial order is a binary relation < that is irreflexive and transitive. Any strict partial order < has an associated partial order ≤ defined by a ≤ b if a < b or a = b. Any preorder ≤ has an associated strict partial order defined by a < b if a ≤ b but b ≤ a. For partial orders ≤, these two ...
Math 7 Scope and Sequence By
... Compare and classify shapes and solids using geometric vocabulary and properties Use coordinate geometry to describe location on a plane Use geometry to model and describe physical world Solves application problems involving estimation and measurement ...
... Compare and classify shapes and solids using geometric vocabulary and properties Use coordinate geometry to describe location on a plane Use geometry to model and describe physical world Solves application problems involving estimation and measurement ...
possible-worlds semantics for modal notions conceived as predicates
... We shall stick as closely as possible to the common possible-worlds semantics for operators, and we shall not apply tricks to avoid paradoxes. That is, we shall use classical logic throughout, in contrast to other authors who have applied the techniques known from the theory of truth, e.g., partial ...
... We shall stick as closely as possible to the common possible-worlds semantics for operators, and we shall not apply tricks to avoid paradoxes. That is, we shall use classical logic throughout, in contrast to other authors who have applied the techniques known from the theory of truth, e.g., partial ...
Counting degenerate polynomials of fixed degree and bounded height
... This gives the next upper bounds for Dn (H) (and Dn∗ (H)), which (although they are not sharp for n > 2) will be useful afterwards. Proposition 2.7. For each integer n ≥ 2, we have Dn (H) = O(H n−1) and Dn∗ (H) = O(H n ). Proof. Notice that, for fixed n ≥ 2, there are only finitely many roots of uni ...
... This gives the next upper bounds for Dn (H) (and Dn∗ (H)), which (although they are not sharp for n > 2) will be useful afterwards. Proposition 2.7. For each integer n ≥ 2, we have Dn (H) = O(H n−1) and Dn∗ (H) = O(H n ). Proof. Notice that, for fixed n ≥ 2, there are only finitely many roots of uni ...
ON THE LIMIT POINTS OF THE FRACTIONAL PARTS OF
... number if d > 2 and α2 is positive [3]. By a result of Smyth [10] claiming that each circle |z| = r contains at most two conjugates of a Pisot number α, the inequality α2 > |α3 | holds for every strong Pisot number α. Recall that a result of Pisot and Vijayaraghavan (see, e.g., [2]) implies that if ...
... number if d > 2 and α2 is positive [3]. By a result of Smyth [10] claiming that each circle |z| = r contains at most two conjugates of a Pisot number α, the inequality α2 > |α3 | holds for every strong Pisot number α. Recall that a result of Pisot and Vijayaraghavan (see, e.g., [2]) implies that if ...
Primalitv Testing and Jacobi Sums
... favorably with the older tests discussed by Williams [26], In fact, Williams never found a prime number of this size that took more than 20 minutes to prove prime on an Amdahl 470-V7 computer. On the other hand, these older tests are slower for sufficiently large n. It should also be taken into acco ...
... favorably with the older tests discussed by Williams [26], In fact, Williams never found a prime number of this size that took more than 20 minutes to prove prime on an Amdahl 470-V7 computer. On the other hand, these older tests are slower for sufficiently large n. It should also be taken into acco ...
lecture notes in logic - UCLA Department of Mathematics
... τ = (Const, Rel, Funct, arity), where the sets of constant symbols Const, relation symbols Rel, and function symbols Funct have no common members and arity : Rel ∪ Funct → {1, 2, . . . }. A relation or function symbol P is n-ary if arity(P ) = n. We will often assume that these sets of names are fin ...
... τ = (Const, Rel, Funct, arity), where the sets of constant symbols Const, relation symbols Rel, and function symbols Funct have no common members and arity : Rel ∪ Funct → {1, 2, . . . }. A relation or function symbol P is n-ary if arity(P ) = n. We will often assume that these sets of names are fin ...
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.