Analysis and how to solve things (methods)

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In summary, the speaker is struggling with solving exercises for their analysis homework and lacks mathematical maturity. They are seeking methods or guidelines to help them understand the logical steps required to prove something and how to know if their result is valid. They are also struggling with abstract reasoning and understanding the objectives of exercises. An example exercise is given and the speaker discusses how they can solve it by finding values for x that satisfy the inequality. They are looking for a general process to follow when solving analysis problems.
  • #1
gillouche
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Hi !

at the moment, I have big troubles to solve exercices for my analysis homework. This is not a problem with an exercise in particular, it really is about how to solve things. I just started a bachelor's degree in physics and I have no mathematical maturity. I can easily use mathematics to solve physics problem but I have no idea what I am doing with "pure mathematics" (analysis).

I fail to understand the logical steps I have to make to prove anything.

I would like to know if there was some sort of methods/guidelines/anything really, I could follow for some cases that would help me see the steps I have to make to prove something and more importantly, how to know that my result is valid. How to know that I need to do a proof by contradiction or something else ?

I asked the same question to the teaching assistant and I am waiting for an answer but I thought that I could ask other people to have maybe more help.

How to get better at abstract reasoning for unknown solutions ? For example, I understand the proofs in class, I can do them again but I am totally unable to think like that myself. I really don't understand what my objectives are when I read an exercise.

Here is an example :

Let b ∈ R be a fixed real number, solve the inequality x2 ≤ b2. Give a necessary and sufficient condition on a and b so that a2 ≤ b2.

If I solve the inequality I have (x-b)(x+b) <= 0 but after that, I don't know what I need to do.

Thank you.
 
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  • #2
gillouche said:
Hi !

at the moment, I have big troubles to solve exercices for my analysis homework. This is not a problem with an exercise in particular, it really is about how to solve things. I just started a bachelor's degree in physics and I have no mathematical maturity. I can easily use mathematics to solve physics problem but I have no idea what I am doing with "pure mathematics" (analysis).

I fail to understand the logical steps I have to make to prove anything.

I would like to know if there was some sort of methods/guidelines/anything really, I could follow for some cases that would help me see the steps I have to make to prove something and more importantly, how to know that my result is valid. How to know that I need to do a proof by contradiction or something else ?

I asked the same question to the teaching assistant and I am waiting for an answer but I thought that I could ask other people to have maybe more help.

How to get better at abstract reasoning for unknown solutions ? For example, I understand the proofs in class, I can do them again but I am totally unable to think like that myself. I really don't understand what my objectives are when I read an exercise.

Here is an example :
If I solve the inequality I have (x-b)(x+b) <= 0 but after that, I don't know what I need to do.

Well, ask yourself: How do I make the product (x-b)(x+b) = 0? This is a simple question. All you need is a knowledge of arithmetic to answer it.
 
  • #3
x must be equal to b or -b for the inequality to be equal to zero. But what about the inequality lower than 0? I can try to guess values of x to get a result lower than 0 like if x < b, then I get a product < 0.

So the possible values for x to to satisfy this inequality are b, -b and x < b. All those values satisfy the x2 <= b2. Is that a satisfactory answer for the first part ?

After that, how do I answer the rest of the question about a and b ?

This sounds more like a homework problem but I didn't want my post to turn like this. My main problem (I am sure I have others) is not about a specific exercises but I have no process in my head that I can follow and/or apply to analysis problems to solve them.

I am more looking for generalization like if I have an inequality ab >= 0 to prove, I should try to prove ab < 0 and if I get a result which is impossible, I would have proven by contradiction that ab can only be greater or equal than 0. This is an example to help me visualize a process which will help me solve that kind of exercises;
 
  • #4
gillouche said:
x must be equal to b or -b for the inequality to be equal to zero. But what about the inequality lower than 0? I can try to guess values of x to get a result lower than 0 like if x < b, then I get a product < 0.

So the possible values for x to to satisfy this inequality are b, -b and x < b. All those values satisfy the x2 <= b2. Is that a satisfactory answer for the first part ?

After that, how do I answer the rest of the question about a and b ?

This sounds more like a homework problem but I didn't want my post to turn like this. My main problem (I am sure I have others) is not about a specific exercises but I have no process in my head that I can follow and/or apply to analysis problems to solve them.

I am more looking for generalization like if I have an inequality ab >= 0 to prove, I should try to prove ab < 0 and if I get a result which is impossible, I would have proven by contradiction that ab can only be greater or equal than 0. This is an example to help me visualize a process which will help me solve that kind of exercises;
You have to analyze these problems in steps: there are very few which can be solved by using this formula or that. This is the essence of analysis: to analyze the problem.

You want to find out what values of x make this relation, (x-b)(x+b) ≤ 0, true. You know that x = b or x = -b satisfy the equality portion. What happens if x > b? If x > -b?

Sometimes, it comes down to a process of exploring a finite number of possibilities.
 
  • #5
gillouche said:
Hi !

at the moment, I have big troubles to solve exercices for my analysis homework. This is not a problem with an exercise in particular, it really is about how to solve things. I just started a bachelor's degree in physics and I have no mathematical maturity. I can easily use mathematics to solve physics problem but I have no idea what I am doing with "pure mathematics" (analysis).

I fail to understand the logical steps I have to make to prove anything.

I would like to know if there was some sort of methods/guidelines/anything really, I could follow for some cases that would help me see the steps I have to make to prove something and more importantly, how to know that my result is valid. How to know that I need to do a proof by contradiction or something else ?

I asked the same question to the teaching assistant and I am waiting for an answer but I thought that I could ask other people to have maybe more help.

How to get better at abstract reasoning for unknown solutions ? For example, I understand the proofs in class, I can do them again but I am totally unable to think like that myself. I really don't understand what my objectives are when I read an exercise.

Here is an example :
If I solve the inequality I have (x-b)(x+b) <= 0 but after that, I don't know what I need to do.

Thank you.
For a product of two terms to be negative, one must be negative and the other must be positive.
 

1. What is analysis and why is it important?

Analysis is the process of breaking down a complex problem or situation into smaller, more manageable parts in order to gain a better understanding of it. It is important because it allows us to identify the root cause of a problem and develop effective solutions.

2. What are the different methods of analysis?

There are several methods of analysis, including qualitative and quantitative analysis. Qualitative analysis involves examining non-numerical data, such as words or images, to identify patterns and themes. Quantitative analysis involves using numerical data and statistical methods to measure and analyze data.

3. How do I choose the best analysis method for my problem?

The best analysis method depends on the type of problem you are trying to solve and the data available. If the problem is complex and involves a lot of variables, a combination of both qualitative and quantitative analysis may be necessary. It is important to carefully consider the problem and the available resources before deciding on an analysis method.

4. What are the steps involved in solving a problem using analysis?

The first step is to clearly define the problem and gather relevant data. Then, you can use the chosen analysis method to break down the problem and identify patterns or trends. Next, analyze the data and draw conclusions based on the results. Finally, use your findings to develop and implement a solution.

5. What are some tips for effectively using analysis to solve problems?

Some tips for effective analysis include being open-minded, paying attention to details, and being systematic in your approach. It is also important to involve multiple perspectives and consider potential biases. Additionally, regularly reviewing and adjusting your analysis can help ensure that you are on the right track towards finding a solution.

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