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I came across some interesting mistakes in many area of mathematics with my students and do not let me also to tell you for university students level, I would like to know How do i deal with students who make these mistakes :

Mistakes:

1) For ‎$x,y >0 , \log (x+y)=\log x+\log y ‎$

2) For ‎$z\in {C} ,\cos²z+\sin²z \neq 1 ‎$

3) ‎$1 ‎$ is a prime number because ‎$1 ‎$ divides ‎$1 ‎$ and itself

4) For ‎$x, y \in {R},\sqrt{(x²+y²)}=x+y ‎$

5) For ‎ ‎$g(x)\neq 0, \displaystyle \int_{a}^{b}\frac{f(x)}{g(x)}dx=\frac {\int_{a}^{b} f(x)}{\int_{a}^{b}g(x)}dx ‎$

6) ‎${0}^{\infty}‎$ is indeterminate case .

Some Teachers in mathematics believe that the claim 6 is true .

My question here is : How do i deal with students who make these mistakes?

Edit: I edited the question to be clear and according to the below comments

Thank you for any help !!!

zeraoulia rafik
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    As a passing comment, I would suggest wording these a little more carefully for students. "For $x,y \ldots$" is ambiguous --- do you mean "for some $x,y$" or "for all $x,y$"? I can guess which you mean, but seeing this makes me want to ask: Is there a good way to get teachers to pay more attention, and say what they mean rather than mean what they say? – Dave L Renfro Dec 27 '16 at 16:48
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    I thought your question is good, but it was hard to read because of many spelling and grammar mistakes. I tried to edit the grammar and spelling so that it would be easier for people to understand. I didn't understand , "do not let me also to tell you for university students level". Can you explain that?? – Amy B Dec 27 '16 at 19:30
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    (Although I was not the down-voter...) I see an issue with this question as being overly broad. In particular, to believe that there is "a good teaching method" to avoid all of those mistakes. For example, your first listed mistake alone is discussed at length in an earlier post, MESE 926. There is not a panacea for all the ills that you describe; so, I am not sure how to respond to all of these questions... maybe focus on one. – Benjamin Dickman Dec 27 '16 at 19:31
  • @AmyB, sorry for my language , I meant by "do not let me also to tell you for university students level" also students university level believe the titled mistakes are true – zeraoulia rafik Dec 27 '16 at 22:53
  • @DaveLRenfro , I meant for all x,y not for some x, y , they are mixed between log(x.y) and log(x+y) !!!! – zeraoulia rafik Dec 27 '16 at 22:56
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    I voted to close mainly for two reasons. First, as @BenjaminDickman implied above, the question is too broad. Second, and I think, more importantly, it is wrong to ask how to "avoid these mistakes". We might ask how to "deal with these mistakes". That is quite another question, yet too broad. – Amir Asghari Dec 28 '16 at 12:32
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    Not only too broad, but such different kinds of issues. Numbers 1, 4, and 5 are easily dealt with by trying an example. (And maybe #6.) How to get students to do that is a bigger question. #3 is by definition, though. – Sue VanHattum Dec 28 '16 at 14:38
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    I see from the comments that your questions is considered too broad. Would you consider focusing on one type of mistake and asking how to deal with students who make that mistake? If you get help you can then ask another question. – Amy B Dec 28 '16 at 16:59
  • The question may be too broad, but the answer is rather simple: these mistakes are the result of students not being familiar enough with the subject. They simply need to study more. One way of doing that is to let them try to prove or test their statements. Point 3 is a misconception about the definition prime numbers, and for 6 it is possible to reason with (philosophical) induction (00=0,00*0=0, etc) which can be rather persuasive, although not mathematically correct. The others should work fine. – jogco Dec 30 '16 at 11:14

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