---
description: math runway, jephian, nsysu, 林晉宏
tags: talk, learning-together, math-runway
---

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# Common English usage in Mathematics

## 課堂常用句法

### 老師

- You are correct ... but your answer is not quite complete. Please consider ...
- Can someone volunteer to summarize in your own words the main points we just discussed?
- That is all for today---we will see you next class.
- The results seem to demonstrate that ...
- These findings prove that ...
- To be honest, it seems to me that ...

### 學生

- I don't quite understand ... . I am fine with _something that you are okay_ but I got lost when we are talking about _something that you don't understand_. Could you give me some hints?
- Could you go back to ... . Why _something_ is _something_?
- Could you say _something_ again?
- I need some help on _something_.

## 寫證明常用句法

### 令、考慮、假設

- **Let** _condition_. **Then** _implication_.
- **Consider** _condition_. **Then** _implication_.
- **Suppose** _condition_. **Then** _implication_.
- **Assume** _condition_. **Then** _implication_.
- **If** _condition_, **then** _implication_.

:::info
Tips :bulb:

- **令** 或 **考慮** 通常在說一個事實或是設定。
:o: Let $\epsilon > 0$.
:x: Let $\sqrt{2}$ be a rational number.
- **假設** 通常是反證法的或是設定。
:o: Suppose $\sqrt{2}$ can be written as $\frac{a}{b}$ for some integers $a$ and $b$.
:x: Suppose $0$ is a rational number.
- 「if」是連接詞，不會單獨成一個句子
:o: If $x$ is a real number, then $x^2 \geq 0$.
:x: If $x$ is a real number. Then $x^2 \geq 0$.
- 英文標點後空一格，標點前不空格
:::

### 因為、所以

- **Since** _cause_, _effect_.
- **Becuase** _cause_, _effect_.
- **By** _cause_, _effect_.
- _Cause_ **implies** _effect_.
- _Cause_, **so** _effect_.
- _Cause_. **Therefore**, _effect_.
- _Cause_. **Consequently**, _effect_.
- _Cause_. **Hence**, _effect_.
- _Cause_. **Thus**, _effect_.

:::info
Tips :bulb:

- 英文中的因為和所以不會同時出現。
:o: Since $x$ is even, $x+1$ is odd.
:x: Since $x$ is even, so $x+1$ is odd.
- 「because」或是「since」放句首的話子句結束要加逗點。
:o: Since $x$ is even, $x+1$ is odd.
:o: We know $x+1$ is odd since $x$ is even. （通常不會讓數學式放句首，所以會塞一些不改變語意的文字）
:x: Since $x$ is even $x+1$ is odd.
:::

### 論證手法概述

- **It is sufficient/enough to show** ... .
- **Prove by induction on $n$**.
- We will prove the **contrapositive statement**.
- **Suppose, for the purpose of yielding a contradiction,** _hypothesis_.

### 練習

:::success
**Exercise 1 (集合相等)**: Prove that $\{3k + 1: k\in\mathbb{Z}\} = \{3k - 2: k\in\mathbb{Z}\}$.

==_____== $X = \{3k + 1: k\in\mathbb{Z}\}$ and $Y = \{3k - 2: k\in\mathbb{Z}\}$. ==_____== to show that $x\in X$ implies $x\in Y$ and $y\in Y$ implies $y\in X$.

==_____== $x\in X$. ==_____== $x$ can be written as $x = 3k - 1$ for some $k$. ==_____==, $x = 3(k-1) + 2$ ==_____== $x\in Y$.

On the other hand, ==_____== $y\in Y$. ==_Finish the other direction._==
:::

:::success
**Exercise 2 (一對一)**: Determine if the following functions are injective.

1. $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^3$.
2. $g: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^2$.
3. $h: [0,1] \rightarrow \mathbb{R}$ defined by $g(x) = x^2$.

**Claim**: $f$ is injective.
==_____== $a$ and $b$ be numbers in $\mathbb{R}$ such that $a \neq b$. We may ==_____== $a < b$. This ==_____== that $f(a) = a^3 < b^3 = f(b)$, ==_____== $f$ is injective.

**Claim**: $g$ is not injective.
It is ==_____== to find distince $a$ and $b$ in $\mathbb{R}$ such that $g(a) = g(b)$. For example, $g(-1) = g(1)$, ==_____== $g$ is not injective.

**Claim**: $h$ is injective.
==_Finish this case._==
:::

:::success
**Exercise 3 (映成)**: Determine if the following functions are surjective.

1. $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^3$.
2. $g: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = x^2$.
3. $h: [0,1] \rightarrow \mathbb{R}$ defined by $g(x) = x^2$.

**Claim**: $f$ is surjective.
==_____== $y$ be a number in $\mathbb{R}$. ==_____== we may find $x = \sqrt[3]{y}$ such that $f(x) = y$. ==_____==, $f$ is surjective.

**Claim**: $g$ is surjective.
==_Finish this case._==

**Claim**: $h$ is not surjective.
It is ==_____== to find some $y$ in $\mathbb{R}$ such that $y$ cannot be written as $h(x)$ for any $x\in [0,1]$. Observe that $0 \leq x^2 \leq 1$ whenever $0 \leq x \leq 1$, ==_____== we may choose, for example, $y = 100$ so that $y \neq h(x)$ for any $x \in [0,1]$. ==_____==, $h$ is not surjective.
:::

:::success
**Exercise 4 (反證)**: Prove that $\sqrt{2}$ is a rational number.

==_____== $\sqrt{2}$ is a rational number. ==_____== it can be written as $\sqrt{2} = \frac{a}{b}$ for some integers $a$ and $b$ with $\gcd(a,b) = 1$. By taking the square on both sides, we have

$$
2 = \frac{a^2}{b^2},
$$

==_____== $2 b^2 = a^2$. ==_____==, $a$ has to be an even number and we may write it as $a = 2k$. ==_____==, $2b^2 = 4k^2$ and $b^2 = 2k^2$. This again ==_____== $b$ is an even number. However, the fact that $a$ and $b$ are both even numbers violates our assumption $\gcd(a,b) = 1$, ==_____== $\sqrt{2}$ is not a rational number.
:::