MY-ASSIGNMENTEXPERT™可以为您提供 hawaii MATH455 Mathematical Logic数理逻辑的代写代考和辅导服务！
这是夏威夷大学 数理逻辑的代写成功案例。
MATH455课程简介
A system of first order logic. Formal notions of well-formed formula, proof, and derivability. Semantic notions of model, truth, and validity. Completeness theorem. Pre: 321 or graduate standing in a related field or consent. Recommended: 454.Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
Prerequisites
Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert’s program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.
MATH455 Mathematical Logic HELP（EXAM HELP， ONLINE TUTOR）
Assume the following predicates:
$H x: x$ is a human
$C x: x$ is a car
$T x: x$ is a truck
Dxy: $x$ drives $y$
Write formulas representing the obvious assumptions: no human is a car, no car is a truck, humans exist, cars exist, only humans drive, only cars and trucks are driven, etc.
Solution.
No human is a car. $\neg \exists x(H x \& C x)$
No car is a truck. $\neg \exists x(C x \& T x)$
Humans exist. $\exists x H x$
Cars exist. $\exists x C x$
Only humans drive. $\forall x((\exists y D x y) \Rightarrow H x)$
Only cars and trucks are driven. $\forall x((\exists y D y x) \Rightarrow(C x \vee T x))$
Some humans drive. $\exists x$ (Hx\& $\exists y D x y)$
Some humans do not drive. $\exists x(H x \& \neg \exists y D x y)$
Some cars are driven. $\exists x(C x \& \exists y D y x)$. etc, etc.
Write formulas representing the following statements:
(a) Everybody drives a car or a truck.
Solution. $\forall x(H x \Rightarrow \exists y(D x y \&(C y \vee T y)))$
(b) Some people drive both.
Solution. $\exists x(H x \& \exists y \exists z(D x y \& C y \& D x z \& T z))$
(c) Some people don’t drive either.
Solution. $\exists x(H x \& \neg \exists y(D x y \&(C y \vee T y)))$
(d) Nobody drives both.
Solution. $\neg \exists x(H x \& \exists y \exists z(D x y \& D x z \& C y \& T z))$
Assume in addition the following predicate:
Ixy: $x$ is identical to $y$
Write formulas representing the following statements:
(a) Every car has at most one driver.
Solution. $\forall x(C x \Rightarrow \forall y \forall z((D y x \& D z x) \Rightarrow I y z))$
(b) Every truck has exactly two drivers.
Solution. $\forall x(T x \Rightarrow \exists y \exists z((\neg I y z) \& D y x \& D z x \& \forall w(D w x \Rightarrow(I w y \vee I w z))))$
(c) Everybody drives exactly one vehicle (car or truck).
Solution. $\forall x(H x \Rightarrow \exists y(D x y \&(C y \vee T y) \& \forall z((D x z \&(C z \vee T z)) \Rightarrow I y z)))$
Assume the following predicates:
$$
\begin{aligned}
& \text { Ixy: } x=y \
& \text { Pxyz: } x \cdot y=z
\end{aligned}
$$
Write formulas representing the axioms for a group: axioms for equality, existence and uniqueness of products, associative law, existence of an identity element, existence of inverses.
Solution.
(a) equality axioms:
i. $\forall x \operatorname{Ixx}$ [reflexivity]
ii. $\forall x \forall y(I x y \Leftrightarrow I y x)$ [symmetry]
iii. $\forall x \forall y \forall z((I x y \& I y z) \Rightarrow I x z)$ [transitivity]
iv. $\forall x \forall x^{\prime} \forall y \forall y^{\prime} \forall z \forall z^{\prime}\left(\left(I x x^{\prime} \& I y y^{\prime} \& I z z^{\prime}\right) \Rightarrow\left(P x y z \Leftrightarrow P x^{\prime} y^{\prime} z^{\prime}\right)\right)$ [congruence with respect to $P$ ]
(b) existence and uniqueness of products:
i. $\forall x \forall y \exists z$ Pxyz [existence]
ii. $\forall x \forall y \forall z \forall w(($ P xyz \& Pxyw) $\Rightarrow I z w)$ [uniqueness]
(c) associative law:
$$
\forall x \forall y \forall z \exists u \exists v \exists w \text { (Pxyu \& Pyzv \& Puzw \& Pxvw) }
$$
(d) existence of identity element: $\exists u \forall x($ Puxx \& Pxux)
(e) existence of inverses: $\exists u \forall x \exists y($ Puxx \& Pxux\& Pxyu \& Pyxu)
MY-ASSIGNMENTEXPERT™可以为您提供 HAWAII MATH455 MATHEMATICAL LOGIC数理逻辑的代写代考和辅导服务！
