Real Analysis Note 1.1. Logics

此篇笔记是【实分析笔记】系列的第1篇，记录了实分析和微积分之间的联系和区别，并介绍了数学分析证明所需要的逻辑符号和定义。

What is Real Analysis?

Real analysis is what mathematicians would call the rigorous version of calculus.
数学家把实分析称为严格的微积分。

• Being “rigorous“ means that every step we take and every formula we use must be proved.
  “严格”意味着我们进行的每一步以及使用的每一个公式都必须得到证明
• If we start from a set of basic assumptions, called axioms or postulates, we can always get to where we are now by taking one justified step after another.
  如果从一组称为公理或假说的基本假设出发，那么我们总是可以通过一个又一个合理的步骤得到最终想要的结论．

In calculus, we might have proved some important results, but we also took many formulas for granted.
在微积分中，我们可能已经证明了一些重要结论，但其中有很多公式被认为是显而易见便直接使用

• What exactly are limits?
  究竟什么是极限?
• How do you really know when an infinite sum “converges” to one number?
  如何确定什么时候无穷和会“收敛”到一个数？
• In an introductory real analysis course, we are reintroduced to concepts we’ve seen before – continuity, differentiability, and so on
  每一门实分析的入门课都会重新介绍连续性、可微性等这些曾见过的概念
  • but this time, their foundations will be clearly laid
    但这一次，我们要弄清楚这些概念的本质
  • And when you are done, we will have basically proven that calculus works.
    当弄清楚这些之后，我们基本上就证明了微积分的合理性

How to Study Real Analysis

Real analysis is typically the first course in a pure math curriculum,
实分析通常是纯数专业的第一门课程，

• because it introduces us to the important ideas and methodologies of pure math in the context of material we are already familiar with.
  因为它在熟悉的材料背景下向我们介绍纯数学的重要思想和方法．

• Once we are able to be rigorous with familiar ideas, we can apply that way of thinking to unfamiliar territory.
  一旦严格地掌握了那些熟悉的概念，我们就可以把这种思维方式应用到不熟悉的领域里

• At the core of real analysis is the question: “how do we expand our intuition for certain concepts—such as sums—to work in the infinite cases?”
  实分析的核心问题是：“如何把某些基于直觉的概念（比如数的和）推广到无限的情形？”
• Puzzles such as infinite sums cannot be properly understood without being rigorous.
  如果不进行严格论证，我们就无法理解像无穷和这样的难题
• Thus, we must build our hard-core proving skills to apply them to these new, more interesting problems.
  因此，我们必须掌握那些核心的证明技巧，进而将它们应用到那些更有趣的新问题上

The way we learn real analysis is NOT by memorizing formulas or algorithms and plugging things in，
实分析并不适合通过记忆公式和算法并将已知条件代入的学习方法，

• Rather, we need to read and reread definitions and proofs until we understand the larger concepts at work, so we can apply those concepts in our own proofs.
  相反，我们要反复地阅读定义和证明，直到理解了更宽泛的概念，这样才能把这些概念应用到自己的证明中
• The best way to get good at this is to take your time; read slowly, write slowly, and think carefully.
  要做到这一点，最好的方法就是慢慢来：慢慢读，慢慢写，并仔细思考．

1. Statements

The language of mathematics consists primarily of declarative sentences.
数学的语言主要由陈述句构成。

• If a sentence can be classified as true or false, it is called a statement.
  如果一个句子可以被归类为真或假，那么它被称为命题。
• The truth or falsity of a statement is known as its truth value.
  命题的真或假称为它的真值。
• For a sentence to be a statement, it is not necessary that we actually know whether it is true or false, but it must clearly be the case that it is one or the other.
  一个句子要成为命题，并不需要我们实际知道它是真还是假，但它必须明确是其中之一。

Example

Consider the following sentences.
请考虑以下句子。
(a) Two plus two equals four.
二加二等于四。
(b) Every continuous function is differentiable.
每个连续函数都是可微的。
(c) $x^2 - 5x + 6 = 0$
(d) A circle is the only convex set in the plane that has the same width in each direction.
圆是平面中唯一在每个方向上具有相同宽度的凸集。
(e) Every even number greater than 2 is the sum of two primes.
每个大于 2 的偶数都可以表示为两个素数之和。

Solution:

• Sentences (a) and (b) are statements since (a) is true and (b) is false.
  句子 (a) 和 (b) 是命题，因为 (a) 是真的，而 (b) 是假的。

• Sentence (c), on the other hand, is true for some $x$ and false for others.
  另一方面，句子 (c) 对某些 $x$ 是真的，而对其他 $x$ 是假的。
  • If we have a particular context in mind, then (c) will be a statement.
    如果我们有一个特定的语境，那么 (c) 就会是一个命题。
  • In next section we shall see how to remove this ambiguity.
    在下一节中，我们将看到如何消除这种歧义。
• Sentences (d) and (e) are more difficult.
  句子 (d) 和 (e) 更复杂一些。
  • You may or may not know whether they are true or false, but it is certain that each sentence must be one or the other.
    你可能知道，也可能不知道它们是真还是假，但可以肯定的是，每个句子一定是其中之一。
  • Thus (d) and (e) are both statements.
    因此，(d) 和 (e) 都是命题。

2. Sentential Connectives

In studying mathematical logic we shall not be concerned with the truth value of any particular simple statement.
在研究数学逻辑时，我们不会关注任何特定简单命题的真值。

• To be a statement, it must be either true or false (and not both), but it is immaterial which condition applies.
  作为一个命题，它必须是真或假（二者不可兼得），但具体适用哪种情况并不重要。
• What will be important is how the truth value of a compound statement is determined by the truth values of its simpler parts.
  重要的是复合命题的真值是如何由其更简单部分的真值决定的。
• In everyday English conversation we have a variety of ways to change or combine statements.
  在日常英语对话中，我们有多种方式来改变或组合命题。

A simple statement like
一个简单的命题，例如

\(\text{It is windy.}\)
can be negated to form the statement
可以通过取反形成命题

\[\text{It is }{\color{blue}\text{ NOT } } \text{windy.}\]

The compound statement
复合命题

\[\text{It is windy}{\color{blue}\text{ AND }}\text{the waves are high.}\]

is made up of two parts: “It is windy” and “The waves are high.”

• These two parts can also be combined in other ways. For example,
  这两个部分也可以通过其他方式组合。例如：

\[\text{It is windy}{\color{blue}\text{ OR }}\text{the waves are high.}\] \[{\color{blue}\text{If }} \text{ it is windy,}{\color{blue}\text{ then }}\text{the waves are high.}\] \[\text{It is windy}{\color{blue}\text{ if and only if }}\text{the waves are high.}\]

The words in blue above ( not, and, or, if . . . then, if and only if ) are called sentential connectives.
上面蓝色的词被称为命题连接词。

• Their use in mathematical writing is similar to (but not identical with) their everyday usage.
  它们在数学写作中的使用与日常用法相似，但并不完全相同。

• To remove any possible ambiguity, we shall look carefully at each and specify its precise mathematical meaning.
  为了消除任何可能的歧义，我们将仔细研究每一个，并明确其精确的数学含义。

2.1. Negation of a Statement

Let $\color{blue} p $ stand for a given statement.
设 $\color{blue} p $ 表示一个给定的命题。

• Then $\color{blue} \sim p $ (read “not $\color{blue} p $”) represents the logical opposite (negation) of $\color{blue} p $.
  那么 $\color{blue} \sim p $（读作“非 $\color{blue} p $”）表示 $\color{blue} p $ 的逻辑相反（否定）。

• When $\color{blue} p $ is true, $\color{blue} \sim p $ is false; when $\color{blue} p $ is false, $\color{blue} \sim p $ is true.
  当 $\color{blue} p $ 为真时，$\color{blue} \sim p $ 为假；当 $\color{blue} p $ 为假时，$ \color{blue}\sim p $ 为真。

This can be summarized in a truth table:
这可以用一个真值表来总结：

• where ( T ) stands for true and ( F ) stands for false.
  其中，( T ) 表示真，( F ) 表示假。

  2.2. Conjunction of Statements (AND)

  The connective and is used in logic in the same way as it is in ordinary language.
  逻辑中的连接词 and（与）与日常语言中的用法相同。

• If $p$ and $q$ are statements, then the statement $p$ and $q$ (called the conjunction of $p$ and $q$ and denoted by $p \land q$) is true only when both $p$ and $q$ are true,
  如果 $p$ 和 $q$ 是命题，那么命题 $p$ and $q$（称为 $p$ 和 $q$ 的合取，记作 $p \land q$）只有当 $p$ 和 $q$ 均为真时才为真，
• and it is false otherwise.
  在其他情况下为假。

2.3. Disjunction of Statements (OR)

The connective or is used to form a compound statement known as a disjunction.
逻辑中的连接词 or（或）用于形成一个称为析取的复合命题。

In common English, the word or can have two meanings.
在日常英语中，单词“or”（或）可以有两种含义。

In the sentence
在下面的句子中：

\[\text{We'll paint our house yellow or green}\]

the intended meaning is yellow or green, but not both.
这里的意思是黄色或绿色，但不能两者都有。

• This is known as the exclusive meaning of or.
  这被称为“或”的排他性含义。

On the other hand, in the sentence
另一方面，在下面的句子中：

\[\text{Do you want cake or ice cream for dessert?}\]

the intended meaning may include the possibility of having both.
这里的意思可能包括两者都要的可能性。

• This inclusive meaning is the only way the word or is used in logic.
  这种包容性的含义是逻辑中使用 or 的唯一方式。
• Thus, we have the following truth table:

2.4. Conditional Statement (Implication)

A statement of the form

\[\text{ If } \ p,\text{ then } \ q.\]

is called an implication or a conditional statement.
被称为蕴涵命题或条件命题。

• The if-statement $p$ in the implication is called the antecedent and the then-statement $q$ is called the consequent.
  蕴涵命题中“如果”的部分 $p$ 称为前件，“那么”的部分 $q$ 称为后件。

To decide on an appropriate truth table for implication, let us consider the following sentence:
为了确定蕴涵命题的合适真值表，让我们考虑以下句子：

• “If it stops raining by Saturday, then I will go to the football game.“
  “如果周六之前雨停了，那么我会去看足球比赛。”
• If a friend made a statement like this, under what circumstances could you call him a liar?
  如果朋友说了这样的话，在什么情况下你可以说他撒谎了？
• Certainly, if the rain stops and he doesn’t go, then he did not tell the truth.
  显然，如果雨停了但他没有去，那么他就没有说实话。
• But what if the rain doesn’t stop?
  但如果雨没停呢？
• He hasn’t said what he will do then, so whether he goes or not, either is all right.
  他并没有说明这种情况下他会做什么，所以无论他是否去，都没有问题。
  • An implication will be called false only when the antecedent is true and the consequent is false.
    蕴涵命题仅在前件为真而后件为假时被称为假命题。
• If we denote the implication “if $p$, then $q$” by $p \Rightarrow q$, we obtain the following table:
  如果我们用 $p \Rightarrow q$ 表示蕴涵命题“如果 $p$，那么 $q$”，我们可以得到以下真值表：

It is important to recognize that in mathematical writing the conditional statement can be disguised in several equivalent forms.
需要注意的是，在数学写作中，条件命题可以以多种等价形式表达。

• Thus the following expressions all mean exactly the same thing:
  因此，以下表达形式完全等价：

  • If $p$, then $q$.

  • $p$ implies $q$.
  • $p$ only if $q$.
  • $q$, if $p$.
  • $q$ provided $p$.

  • $q$ whenever $p$.

  • $q$ is a necessary condition for $p$.
  • $p$ is a sufficient condition for $q$.

Visualize an Implication

One way to visualize an implication $R \Rightarrow S$ is to picture two sets R and S, with R inside S.
一种将条件命题 $R \Rightarrow S$ 可视化的方法是将其想象为两个集合 R 和 S，其中 R 包含在 S 中。

• Objects that are Round are in set R, and Objects that are Solid are in set S.
  圆形的物体属于集合 R，实心的物体属于集合 S。

We see that the relationship between R and S in Figure above can be stated in several equivalent ways:
我们可以看到，上图 中 R 和 S 的关系可以用多种等价的方式表述：

• If an object is Round (R), then it is solid (S).
  如果一个物体是圆形的 (R)，那么它是实心的 (S)。
• An object is solid (S) whenever it is round (R).
  每当一个物体是圆形的 (R)，它就是实心的 (S)。
• An object is solid (S) provided that it is round (R).
  如果一个物体是圆形的 (R)，那么它是实心的 (S)。

• Being round (R) is a sufficient condition for an object to be solid (S).
  圆形 (R) 是物体为实心 (S) 的充分条件。
  (It is sufficient to know that an object is round to conclude that it is solid.)
  （知道一个物体是圆形的足以推断它是实心的。）

• Being solid (S) is a necessary condition for an object to be round (R).
  实心 (S) 是物体为圆形 (R) 的必要条件。
  (It is necessary for an item to be solid in order for it to be round.)
  （物体必须是实心的才能是圆形的。）

2.5. Biconditional

The statement
命题

\[p \text{ if and only if } q\]

is the conjunction of the two conditional statements $p \Rightarrow q$ and $q \Rightarrow p$.
是两个条件命题 $p \Rightarrow q$ 和 $q \Rightarrow p$ 的合取。

• A statement in this form is called a biconditional and is denoted by
  这种形式的命题称为双条件命题，记作

\[p \Leftrightarrow q\]

• In written form, the abbreviation “iff” is sometimes used instead of “if and only if.”
  在书面表达中，有时使用缩写“iff”来代替“if and only if”。

• Thus we see that $p \Leftrightarrow q$ is true precisely when p and q have the same truth values.
  当p和q两个命题的真值相同时，双条件的真值为真，否则为假。

3. Tautologies and Contradictions

• Tautologies and Contradictions are often important in mathematical reasoning.
  永真命题和矛盾命题在数学推理中往往很重要

  Definition

  A compound statement is called a tautology if it is always True for ALL truth value assignments for its component statements.
  给定一个复合命题，如果对其分量命题无论做怎样的赋值，其对应的真值永远是真，则称该命题为永真命题

• If a compound statement is False for ALL truth value assignments, then it is called a contradiction
  如果无论怎样赋值其真值永远为假，那么这个复合命题为矛盾命题。

Example

• Because $p \ \lor \sim p$ is always True for ALL truth value assignments, it is a tautology.
• Because $p \ \land \sim p$ is always False for ALL truth value assignments, it is a contradiction.

  4. Logical Equivalences

  An important type of step used in a mathematical argument is the replacement of a statement with another statement with the same truth value.
  数学证明中使用的一个重要步骤就是用真值相同的一个命题替换另一个命题。

• Because of this, methods that produce statements with the same truth value as a given compound statement are used extensively in the construction of mathematical arguments.
  因此，从给定的复合命题生成具有相同真值命题的方法广泛使用用数学证明的构造。

Definition

The compound statements $p$ and $q$ are called logically equivalent if the biconditional statement $p \Leftrightarrow q$ is a tautology.
如果 $p \Leftrightarrow q$ 是永真命题，则复合命题 $p$ 和 $q$ 是逻辑等价的。

• Or, Compound statements that have the same truth values in all possible cases are called logically equivalent.
  或者可以说，在所有可能的情况下都有相同真值的两个复合命题互为逻辑等价

Example

• Since The truth table for the biconditional can be obtained by analyzing the compound statement $(p \Rightarrow q) \land (q \Rightarrow p)$ a step at a time.
  因为双条件命题的真值表可以通过逐步分析复合命题 $(p \Rightarrow q) \land (q \Rightarrow p)$ 得到。

• Then we can say $(p \Rightarrow q) \land (q \Rightarrow p)$ and $p \Leftrightarrow q$ are logically equivalence
  因此我们可以说这两个复合命题逻辑等价

  Resources

[1] Steven R. Lay, Analysis with an Introduction to Proof, 5th edition (Pearson, 2012)

[2] Raffi Grinberg, The real analysis lifesaver (Princeton University Press, 2012)