A1-A4 - Tufts
... First show that cos 4 is a constructible number (that is, given a unit length, this length can be constructed with straightedge and compass). Then, using the double-angle formula cos(2θ) = 2 cos2 θ − 1, explain how to use that length to construct cos π8 . Repeating this logic, conclude that π can be ...
... First show that cos 4 is a constructible number (that is, given a unit length, this length can be constructed with straightedge and compass). Then, using the double-angle formula cos(2θ) = 2 cos2 θ − 1, explain how to use that length to construct cos π8 . Repeating this logic, conclude that π can be ...
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
... I | L = I. This means that we have to define cI 0 for all c ∈ C. By the definition, cI 0 ∈ M , so this also means that we have to assign the elements of M to all constants c ∈ C in such a way that the resulting expansion is a model for all sentences from SHenkin . The quantifier axioms Q1, Q2 are fi ...
... I | L = I. This means that we have to define cI 0 for all c ∈ C. By the definition, cI 0 ∈ M , so this also means that we have to assign the elements of M to all constants c ∈ C in such a way that the resulting expansion is a model for all sentences from SHenkin . The quantifier axioms Q1, Q2 are fi ...
Chapter 2 Notes Niven – RHS Fall 12-13
... An argument that follows deductive reasoning is called a proof. A conjecture that has been proven is called a theorem. A proof transforms a conjecture into a theorem. Postulates and theorems are often written in conditional form. Remember that postulates and theorems are different from definitions a ...
... An argument that follows deductive reasoning is called a proof. A conjecture that has been proven is called a theorem. A proof transforms a conjecture into a theorem. Postulates and theorems are often written in conditional form. Remember that postulates and theorems are different from definitions a ...
Logic - Decision Procedures
... (3) I have not filed any of them that I can read; (4) None of them, that are written on one sheet, are undated; (5) All of them, that are not crossed, are in black ink; (6) All of them, written by Brown, begin with "Dear Sir"; (7) All of them, written on blue paper, are filed; (8) None of them, writ ...
... (3) I have not filed any of them that I can read; (4) None of them, that are written on one sheet, are undated; (5) All of them, that are not crossed, are in black ink; (6) All of them, written by Brown, begin with "Dear Sir"; (7) All of them, written on blue paper, are filed; (8) None of them, writ ...
A simple proof of Parsons` theorem
... In fact, for x = tj (c, d1 , . . . , dj−1 ) take y = dj and use the fact that ¬ϕ is a universal formula and, therefore, downward absolute between M and M∗ . We have restricted the statement of the theorem to single variables u, x and y in order to make the proof more readable. It is clear, however ...
... In fact, for x = tj (c, d1 , . . . , dj−1 ) take y = dj and use the fact that ¬ϕ is a universal formula and, therefore, downward absolute between M and M∗ . We have restricted the statement of the theorem to single variables u, x and y in order to make the proof more readable. It is clear, however ...
Notes and exercises on First Order Logic
... interpretation (meaning) in the structure U. This distinction is vital for a proper understanding of the semantics of FOL, and also clarifies many ambiguities of mathematics, where the same symbol is used in different contexts. In practice we often omit the superscript when it is clear what the inte ...
... interpretation (meaning) in the structure U. This distinction is vital for a proper understanding of the semantics of FOL, and also clarifies many ambiguities of mathematics, where the same symbol is used in different contexts. In practice we often omit the superscript when it is clear what the inte ...
CPS130, Lecture 1: Introduction to Algorithms
... Suppose there exists a set S = {k | p´(k) }. Then S has a least element m0 >1 since p(1) is true. By construction of S, we have p(k) is true for all k < m0, including m0-1. Therefore by the second part of the hypothesis for either (I1) or (I2), p(m0) must be true. This is a contradiction ( p(m0 ) ...
... Suppose there exists a set S = {k | p´(k) }. Then S has a least element m0 >1 since p(1) is true. By construction of S, we have p(k) is true for all k < m0, including m0-1. Therefore by the second part of the hypothesis for either (I1) or (I2), p(m0) must be true. This is a contradiction ( p(m0 ) ...
Notes on Classical Propositional Logic
... a truth table (though a proof may be hard to find). In this section we set up the basics of the axiomatic approach, then in subsequent sections we work towards soundness and completeness (and explain what these terms mean). To have an axiom system two things are needed: we need some class of formula ...
... a truth table (though a proof may be hard to find). In this section we set up the basics of the axiomatic approach, then in subsequent sections we work towards soundness and completeness (and explain what these terms mean). To have an axiom system two things are needed: we need some class of formula ...
1 Chapter III Set Theory as a Theory of First Order Predicate Logic
... members;. In fact, it seems reasonable to hold that we can form not only such sets, but also sets which consist partly of subsets of A and partly of members of A; the sets which have only individuals as members and those which have only sets of individuals as members are special cases of this more g ...
... members;. In fact, it seems reasonable to hold that we can form not only such sets, but also sets which consist partly of subsets of A and partly of members of A; the sets which have only individuals as members and those which have only sets of individuals as members are special cases of this more g ...
Chapter 4, Mathematics
... ‘algorithm’. For example the standard procedures for addition, subtraction and multiplication are all algorithms. In logical theory ‘decision procedure’ is equivalent to ‘algorithm’. In cookery a reliable recipe is an algorithm for producing the soup, cake, stew or whatever it is that it tells us ho ...
... ‘algorithm’. For example the standard procedures for addition, subtraction and multiplication are all algorithms. In logical theory ‘decision procedure’ is equivalent to ‘algorithm’. In cookery a reliable recipe is an algorithm for producing the soup, cake, stew or whatever it is that it tells us ho ...
ch1_Logic_and_proofs
... Axioms An axiom is a proposition accepted as true without proof within the mathematical system. There are many examples of axioms in mathematics: ...
... Axioms An axiom is a proposition accepted as true without proof within the mathematical system. There are many examples of axioms in mathematics: ...
Infinitistic Rules of Proof and Their Semantics
... (every non-empty analytical family of unary functions has an analytical element} holds, which is known to be independent from the axioms of set theory. 4. Searching a satisfactory syntactical ,8-rule. It seems that the question raised by Mostowski in [4] about the existence of a syntactical ,8-rule ...
... (every non-empty analytical family of unary functions has an analytical element} holds, which is known to be independent from the axioms of set theory. 4. Searching a satisfactory syntactical ,8-rule. It seems that the question raised by Mostowski in [4] about the existence of a syntactical ,8-rule ...
Hilbert Calculus
... Proof: Assume S ⊢ F → G. Then S ∪ {F } ⊢ F → G. Using S ∪ {F } ⊢ F and Modus Ponens we get S ∪ {F } ⊢ G. Assume S ∪ {F } ⊢ G. Proof by induction on the derivation (length): Axiom/Hypothesis: G is instance of an axiom or G ∈ S ∪ {F }. If F = G use example of derivation to prove S ⊢ F → F . Otherwise ...
... Proof: Assume S ⊢ F → G. Then S ∪ {F } ⊢ F → G. Using S ∪ {F } ⊢ F and Modus Ponens we get S ∪ {F } ⊢ G. Assume S ∪ {F } ⊢ G. Proof by induction on the derivation (length): Axiom/Hypothesis: G is instance of an axiom or G ∈ S ∪ {F }. If F = G use example of derivation to prove S ⊢ F → F . Otherwise ...
