# Cheat Sheet to Write Mathematics in English for Chinese

27 Sep 2015The following is by no means ‘the’ introduction on how to write mathematical proofs in English – English is my third tongue, therefore I do not master it very well myself. Just take the following as building blocks so that I can follow your proofs. After a while, you may improve by looking at proof arguments in professional English textbooks – I am not a reference on that topic. That’s also why I recommend you to read only textbooks in English! At the beginning I accept – even if I hate it – the use of logical symbols such as \(\Rightarrow\), \(\Leftrightarrow\), \(\forall\), \(\exists\), etc. in text. Try however to replace them by their English counterparts, it is nicer to read.*

## Logic:

- Assertion: 断言
- Hypothesis: 假设
- True, False: 真，假
- \(A\Rightarrow B\): “\(A\) implies \(B\)” \(A\) 推出 \(B\) or “From \(A\) follows \(B\)” 若 \(A\) 则 \(B\), or “If \(A\) (is true), then (so is) \(B\).” 若 \(A\) 为真则 \(B\) 为真
- \(A\Leftrightarrow B\): “\(A\) is equivalent to \(B\)” \(A\) 等价于 \(B\) or “\(A\) holds if, and only if, \(B\) holds”. \(A\) 成立当且仅当 \(B\) 成立
- Element: 元素
- Set: 集
- Collection: 类
- \(x\in A\): “\(x\) is in \(A\)”. \(x\) 属于 \(A\)
- \(A\subseteq B\): “\(A\) is a subset of \(B\)”. \(A\) 包含于 \(B\)
- \(\forall x\): “For all \(x\)” (or “For every \(x\)”). 对任意 \(x\)
- \(\exists x\): “For some \(x\)”, or (there exists \(x\)). 存在 \(x\)

## Basic math vocabulary:

- \(x\geq y\): \(x\) is greater than \(y\), or \(y\) is less than \(x\). \(x\) 不小于 \(y\)
- \(x\geq 0\): \(x\) is positive, and \(-x\) is negative. \(x\)不小于\(0\)
- \(x>y\) or \(x>0\): \(x\) is strictly greater than \(y\), or \(x\) is strictly positive. \(x\)严格大于\(y\) \(x\)严格大于\(0\)
- \(\mathbb{N}\): natural numbers (自然数) sometimes called strictly positive integers (正整数).
- \(\mathbb{Z}\): integers 整数
- \(\mathbb{Q}\): rational numbers 有理数
- \(\mathbb{R}\): real numbers 实数
- \(f:X\to Y\): function \(f\) from the domain \(X\) to the codomain \(Y\). \(f\) 是从定义域 \(X\) 映射到值域 \(Y\) 的函数
- \(f(A)\) and \(f^{-1}(B)\): image of \(A\subseteq X\) and preimage of \(B\subseteq Y\) under \(f\). \(A\) 在映射\(f\)下的像，\(B\)的原像
- \(f(x)\leq f(y)\) for \(x\leq y\): \(f\) is increasing. 单调不减的
- \(f(x)\geq f(y)\) for \(x\leq y\): \(f\) is decreasing. 单调不增的
- \(f\) either increasing or decreasing: \(f\) is monotone. \(f\) 是单调的
- Finite family 有限族
- Countable family 可数族

## Writing proofs:

- We show that \(A\) and \(B\) implies \(C\). 我们说明A 和 B 推出C
- Under the assumption [of the theorem/proposition/exercise], it holds [this/that].根据定理的假设，我们可以得出
- Suppose/assume that [something] holds.假设…成立
- By contradiction, suppose that [something] holds. (show that it can not be true).由于矛盾，说明…成立
- If [something] holds, then it follows that … 如果…成立则说明…
- Since [something] holds, it follows that… 因为…成立，我们有…
- Hence,… 由于
- [something] yields [something] …表明…
- However, [something is true], therefore, [another thing] holds. …成立，因此，…成立
- Thus,… 那么
- This completes the proof. (or CQFD.) 证明完成

## Exemplary sentences of the lecture:

- Let \(x \in \mathbb{R}\) be such that… 令x属于实数集则有
- Let \((x_n)\) be a sequence of elements in \(A\) such that… 令\((x_n)\)是A中的一列元素
- Let \(f:X\to Y\) be a function such that… 令f是从X映射到Y的函数，那么
- Let \((A_i)\) be a family of subsets of \(\Omega\) such that… 令\((A_i)\)是\((\Omega)\)中的子集族
- Let \((A_n)\) be a countable family of subsets of \(\Omega\) such that…令\((A_i)\)是\((\Omega)\)中的可数子集族
- Since \(\mathcal{F}\) is a $\sigma$-algebra, it follows that \(\cup A_n \in \mathcal{F}\) for every countable family \((A_n)\) of elements in \(\mathcal{F}\). 因为 \(\mathcal{F}\) 是一个 $\sigma$-代数, 对任意由 \(\mathcal{F}\)中的元素\((A_n)\)组成的可数集族可以得到 \(\cup A_n \in \mathcal{F}\)
- Let \((A_k)_{k\leq n}\) be a finite family of subsets of \(\Omega\) such that…令\((A_k)_{k\leq n}\)是\(\Omega\)的可数子集族，那么
- Suppose that \((A_n)\) is a countable family of subsets of \(\Omega\) such that…假设\((A_n)\)是\(\Omega\)的一个可数子集族
- Since the random variable $X$ is measurable, it follows that \(\{X\leq a\}\) is measurable for every \(a \in \mathbb{R}\).由于随机变量X可测，可以得到\(\{X\leq a\}\)对任意\(a \in \mathbb{R}\)可测
- Suppose that the random variable \(X\) is bounded, then it follows in particular that \(X\) is integrable. 假设随机变量X有界，那么特别地，X是可积的

## Example

(推理)

Let \(n\) and \(m\) be two integers.Suppose that \(n\) is even and \(m\) is even.
Then, \(nm\) is also even.

令 \(n\) 和 \(m\) 是两个整数. 假设 \(n\) 是偶数 \(m\) 是偶数. 那么 \(nm\) 也是偶数.

(not nice but okay at the beginning) 证明（不完美的写法）

\(n,m \in \mathbb{Z}\) are even \(\Rightarrow\) \(\exists p,q \in \mathbb{Z}\) such that \(n=2p\) and \(m=2q\) \(\Rightarrow nm=2p2q=2(2pq)=2r\) where \(r:=2pq\in \mathbb{Z}\) \(\Rightarrow\) \(nm\) is even. CQFD.

\(n,m \in \mathbb{Z}\) 是偶数 \(\Rightarrow\) \(\exists p,q \in \mathbb{Z}\) 使得 n=2p 且 m=2q \(\Rightarrow nm=2p2q=2(2pq)=2r其中r=2pq\in \mathbb{Z}\) \(\Rightarrow\) \(mn\) 也是偶数. 证明完成.

(证明)

Let $n$ and $m$ be two even integers.
By definition, it follows that $n$ and $m$ are divisible by two, that is, \(n=2p\) and \(m=2q\) for some integers \(p\) and \(q\).
Hence, \(nm=2p2q=4pq=2(2pq)\).
It follows that \(nm=2r\) where \(r=2pq\) is an integer. Thus, \(nm\) is even, which completes the proof.

令 \(n\) 和 \(m\) 是两个偶数. 根据定义，\(m\) 和 \(n\) 可以被 \(2\) 整除，这说明 \(n=2p\) 且 \(m=2q\) 对某些整数 \(p,q\) 成立. 那么 \(mn=2r\) 其中 \(r=2pq\) 也是整数. 那么 \(nm\) 是偶数，证明完成.

## Comments

- As French, I never use the US/UK terminology “non-decreasing”, “non-increasing”, “non-negative” and “non-positive” for “increasing”, “decreasing”, “positive” and “negative” as defined above. Indeed, I find this way of defining something by negation unnatural and actually only makes sense for total order. So it is a bad habit, and I want you to use the natural definitions as above. If, however, you decide to go for the Yankee way, then stick to it and do not mix.
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