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.
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- 令 或 考慮 通常在說一個事實或是設定。
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Let .
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Let be a rational number.
- 假設 通常是反證法的或是設定。
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Suppose can be written as for some integers and .
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Suppose is a rational number.
- 「if」是連接詞,不會單獨成一個句子
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If is a real number, then .
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If is a real number. Then .
- 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.
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- 英文中的因為和所以不會同時出現。
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Since is even, is odd.
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Since is even, so is odd.
- 「because」或是「since」放句首的話子句結束要加逗點。
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Since is even, is odd.
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We know is odd since 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)
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Since is even is odd.
論證手法概述
- It is sufficient/enough to show … .
- Prove by induction on .
- We will prove the contrapositive statement.
- Suppose, for the purpose of yielding a contradiction, hypothesis.
練習
Exercise 1 (集合相等): Prove that .
_____ and . _____ to show that implies and implies .
_____ . _____ can be written as for some . _____, _____ .
On the other hand, _____ . Finish the other direction.
Exercise 2 (一對一): Determine if the following functions are injective.
- defined by .
- defined by .
- defined by .
Claim: is injective.
_____ and be numbers in such that . We may _____ . This _____ that , _____ is injective.
Claim: is not injective.
It is _____ to find distinct and in such that . For example, , _____ is not injective.
Claim: is injective.
Finish this case.
Exercise 3 (映成): Determine if the following functions are surjective.
- defined by .
- defined by .
- defined by .
Claim: is surjective.
_____ be a number in . _____ we may find such that . _____, is surjective.
Claim: is not surjective.
Finish this case.
Claim: is not surjective.
It is _____ to find some in such that cannot be written as for any . Observe that whenever , _____ we may choose, for example, so that for any . _____, is not surjective.
Exercise 4 (反證): Prove that is an irrational number.
_____ is a rational number. _____ it can be written as for some integers and with . By taking the square on both sides, we have
_____ . _____, has to be an even number and we may write it as . _____, and . This again _____ is an even number. However, the fact that and are both even numbers violates our assumption , _____ is not a rational number.