As a matter of fact 事實上

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Actually, there’s no way he would agree.
The truth is, he wouldn’t agree to it.
As a matter of fact, it’s impossible for him to agree.

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Take/Make a Note/ Notes / (the note for)

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take a note / make a note: 紀錄 n. 筆記 to take or make a note of sth. 記下某事物 to make a note to do sth. 記下要做某事 vt. 注意到; 注意 to note that ... 注意到… it is interesting to note that ... 有意思的是，… I'll just take a note of your name and address. 我記一下你的姓名和地址。 She made a  mental note  of the title. 她 牢牢地記住 了標題。 Make a note to call again next week.. 一定記得下週再打電話 He never takes notes   in class . His new theory  is worthy of   note. This is the note for sth. 這是sth的筆記 Note it on your smartphone.
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even though (儘管) vs even if (即使)

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Even though (儘管) //強調已發生的事實, 用過去式 Even though it was raining, we went for a hike Even though he studied hard, he didn't pass the exam. Even if (即使, 就算是) //假設, 用現在式+未來式 Even if it is true, you shouldn't say anything. I will go for a run even if it rains. Even if we fail, we will learn from our mistakes
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