【离散数学】 MIT 6.042J 笔记 - Lecture 1 Introductions and Proofs

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【离散数学】MIT 6.042J - Fall 2010 - Note for Lecture 1

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\newtheorem{theorem}{Theorem}\title{\huge \bfseries Mathematics for Computer Science, MIT 6.042J - Fall 2010 - Note for Lecture 1}\author{\Large Teddy van Jerry}
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\chead{MIT 6.042J Lecture 1 - Introduction and Proofs}\begin{document}\maketitle{\Large Lecture 1: Introduction and Proofs}\section{Definition of mathematical proof}\begin{definition}A mathematical proof is a verification of a proposition by a chainof logical deductions from a set axioms.\end{definition}\section{Proposition}\subsection{Example 1}$$ 3 + 2 = 5$$This is a true proposition.\subsection{Example 2}$$\forall n \in \mathbb{N}, n^2 + n + 41 \text{ is a prime number.}$$This is a predicate.\begin{definition}Predicate is a proposition whose truth depends on the value of variable($n$).\end{definition}If $n$ is from $0$ to $39$, the proposition is true.However, when $n = 40$, $40^2 + 40 + 41 = 41^2$ is not a prime number. Thus it is false.\subsection{Example 3}$$a^4 + b^4 + c^4 = d^4 \text{ has no integer solutions.}$$This was proposed by Euler in 1767.However, after 218 years, it was proved false:$$\begin{aligned}a & = 95800\text{,}  \\b & = 217519\text{,} \\c & = 414560\text{,} \\d & = 422481\text{.} \\\end{aligned}$$\subsection{Example 4}$$313(x^3 + y^3) = z^3 \text{ has no positive integer solutions.}$$However, the simplest solution has more than 1000 digits.\subsection{Example 5}The famous four color theorem:The regions in any map can be colored in four colorsso adjacent regions have different colors.It was solved by a computer in 1977.\subsection{Example 6}The Riemann hypothesis:$$\forall n \in \mathbb{Z}, n \geqslant 2 \Rightarrow n^2 \geqslant 4$$The symbol $\Rightarrow$ means \emph{imply}.This proposition is true, however changing $\Rightarrow$ into $\Leftarrow$,it is not true for $n \leqslant -2$.A implication $p$ implies $q$ is said to be true if $p$ is false or $q$ is false.To make it clearer, we have the {\bfseries Truth Table}:\begin{table}[htbp]\centering\caption{Truth Table}\begin{tabular}{|m{1cm}<{\centering}|m{1cm}<{\centering}|m{1cm}<{\centering}|m{1cm}<{\centering}|m{1cm}<{\centering}|}\hline$p$ & $q$ & $p \Rightarrow q$ & $p \Leftarrow q$ & $p \Leftrightarrow q$ \\\hline \hlineT & T & T & T & T \\\hlineT & F & F & T & F \\\hlineF & T & T & F & F \\\hlineF & F & T & T & T \\\hline\end{tabular}\end{table}It can be difficult to understand False implies False is True.One example is `If pigs fly, I'm king',because `pigs fly' is never true,we don't even need to consider `I'm king'.\subsection{Why these solutions count}Many solutions may seem far away from today's circumstance,but as time goes on, numbers can go upwards in a dramatic way,so we are obliged to find the solutions as they can be met at any time.\section{Axiom}\subsection{Definition}\begin{definition}Axioms are propositions we assume to be true.\end{definition}The word `axiom' comes from Greek which means `think worthy'.You have to make assumptions in truth like:$$\left.\begin{aligned}a & = b \\b & = c \\\end{aligned}\right\} \Rightarrow a = c$$However, axioms can be contradictory in different contexts, for example:In Euclidean geometry: Given a line $l$ and a point $p$(not on $l$),there is {\bfseries exactly one line} through $p$ parallel to $l$.In Spherical geometry: Given a line $l$ and a point $p$(not on $l$),there is {\bfseries no line} through $p$ parallel to $l$.In Hyperbolic geometry: Given a line $l$ and a point $p$(not on $l$),there is {\bfseries infinitely many lines} through $p$ parallel to $l$.\subsection{Requirements for axioms}Axioms should be:\begin{enumerate}\item consistent\begin{definition}A set of axioms is consistent if no proposition can be proved to be both true and false.\end{definition}\item complete\begin{definition}A set of axioms is complete if they can be used to prove any proposition is either true or false.\end{definition}\end{enumerate}But sadly, it has been proved impossible to achieve both requirements at the same time.\section*{Appendix}~This note is written in \LaTeX.\LaTeX\ code in my blog: \url{https://blog.csdn.net/weixin_50012998/article/details/113484512}\\The webpage of the video: \url{https://www.bilibili.com/video/BV1zt411M7D2?p=1}\end{document}% Copyright 2021 Teddy van Jerry

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【离散数学】 MIT 6.042J 笔记 - Lecture 1 Introductions and Proofs

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