Advanced Functions Solutions Manual

[Josefa Palsgrove]

Maybethe answer to this is probably as vague as "it depends." I'm a graduate student trying to pick up some extra math skills. I'm reading a couple of texts and attempting the exercises but the difficulty/specificity of some exercises is such that I fear somewhat frequently asking for guidance would be burdensome to whoever I ask for help (peers, professors, my advisor, etc.). My preferred system would be to try my hand at a collection of problems for a few days, and check my thinking against a solutions manual (consulting others if I encounter something I really don't understand).

Some authors encourage instructors to contact them for solutions manuals if they are planning on teaching with their book. As a self-studier, should I generally expect a similar offer extends to me, or would it be frowned upon to ask for a solutions manual?

It's true that your request might not be successful, because, as some commenters suggested, the solutions manual may be kept away from students to prevent cheating. It may be hard for them to verify that you are not a student trying to cheat.

The Solutions and Tests Manual for Exploring Creation with Advanced Chemistry, 2nd Edition contains answers to the review questions, practice problems, and extra practice problems, as well as the tests and test solutions. A set of Test Pages is shrink-wrapped with the manual.

The Solutions and Tests for Exploring Creation with Advanced Chemistry, 2nd Edition, contains answers to Study Guide questions, module summaries, test solutions, and a set of student test pages.

Using an instructor's solutions manual to complete your homework assignments is considered cheating. It goes against the academic integrity policy of most institutions and does not promote independent learning.

No, using the solutions manual to check your answers is still considered cheating. It is important to understand the material and come up with your own solutions rather than relying on someone else's work.

No, using the solutions manual to study for exams is also considered cheating. Exams are designed to test your understanding and application of the material, and using the solutions manual does not demonstrate that.

The consequences of using the solutions manual for your assignments can vary, but it can result in a failing grade for the assignment or even the entire course. It can also lead to disciplinary action from your institution, which can have serious consequences for your academic career.

Often times I "feel" as if I can write a proof to an exercise but most of those times I do not feel confident that the proof that I am thinking of is good enough or even correct at all. I can sort of think a proof in my head, but am not confident this is a correct proof.

Ask a more experienced person. IMHO that's really the only option, and one of the reasons for this is that it is very important for a proof to communicate a result and its justification to another person. If the proof is good enough to convince yourself, that's a start, but the real test is whether you can express it in such a way as to convince someone else.

And BTW... the same applies if the textbook does have a solutions manual. Your proof is inevitably going to be different from the one in the book, and it takes a lot of experience and mathematical understanding to decide whether the differences are important or not.

1) This is your teacher's job to check your proofs. He is experienced and trained to read proofs and determine what is acceptable and suitable for your level (is it safe to ignore some minor flaw? How detailed a computation should be?...) Don't hesitate to ask advice on proof writing, including on non required work and self-study.

2) Do a step-by-step verification: are all formulas correct? Are all equivalences really equivalences? In particular, don't be lazy in that step: really check that the equivalences you used are not implications. Also, check carefully all the conditions before applying a theorem: the Alternating Series Test requires a decreasing sequence? check it! Including obvious conditions: if it is obvious, write it in one or two lines. As a teacher, it always upset me when students complain about their grade for an obvious fact they did not bother to state.

4) Finally, math is not divided into two steps, one when you receive a lecture with definitions and proofs from the teacher like a sacred text, and a step where you do homeworks and try to copy the master. To be critical on your own work, you have to be critical on others'work. Efforce yourself to be question the professor's proofs: why did he introduce this? Can we shorten the proof like that? He used a non-intuitive trick in the proof; can we do it without the trick?

Unfortunately, many people I've met don't fully understand the logical structure of proofs, and that impairs their understanding of every topic, so it's the first thing to make sure you can understand perfectly. To give some examples, you must be able to identify precisely what statements have been derived and what have not, and exactly what assumptions form the scope of each statement.

The next part involving justifying each step in the proof requires you to be able to follow rules strictly. Although intuition is a very good helper in finding the proof, it is not so reliable in checking the proof, and we often have to stick to symbolic manipulation according to rules if we want to be certain of its correctness. If we don't really know what exactly is allowed and what is not, it is time to look very closely at the precise rules and the justifications behind them.

The last thing about carelessness can't be avoided without a computer, however, and the only other alternative besides a formal proof assistant like Mizar is to let someone else see your proof, preferably someone who meets the first two criteria above.

In many proofs, the hard thing is to find the way which path to follow; once that is done they are easy. Faced with a problem, most of the time my result is: "I have no clue how to solve this", "I have a start but I'm stuck at some point", or "I have a proof which is correct unless I made a stupid mistake".

If you have no confidence that your proof is correct apart from possible mistakes, then you likely don't have a proof. If you say "A => B because I say so" in your proof, especially if you say "A => B must be true because otherwise my proof doesn't work", then most likely you don't have a proof. If that isn't the case, then most likely it's just a matter of checking your proof for mistakes.

Anyway, there are many exercises, too many to do them all. To learn, it isn't necessary to do all the proofs to the last excruciating detail; it's enough to get to the point where you can say "if I spent another hour or two then my proof would be faultless". Training your brain to get the right ideas so you can find proofs is the important thing. For new results, you want faultless proofs (and to avoid failing a test :-) For exercises, someone has written a faultless proof at some time.

Use a computer with automated proof checking software, also called a proof assistant or interactive theorem prover. Typically you will need to write your proof in a special, machine readable format (be careful for translation/copy errors), but past that point this field is well studied and computer-based proof checking is generally reliable. Wikipedia has a comparison table of different software for this purpose: _assistant . (If this is preferable to, easier than, or even faster than hand-verifying your proof step-by-step is another question.)

In my opinion, Sudoku puzzles are great because they help introduce people to the concept of "proof". Try a really hard Sudoku puzzle. In the middle, you don't know the whole solution yet, but, you may be able to tell that this particular square "must" contain so-and-so number, so you can write it in. You did not merely "try" this number, you wrote it in for sure. How did you do that? Well, you went through some precise argument. That was a "proof".

Assuming you really care about qualifying your proof to your self before you present it to others, I would think the answer is really simple (and a bit annoying :) ); retrace your entire proof, step by step. Although there is a caveat, you shouldn't be actually attempting to solve the problem 'anew'.

I would lay out my assumptions clearly; maybe look through my solution and write down all the rules and tricks I used. Then I would check if any of my assumptions introduce ideas I didnt really think I was using. This can be even harder but, having a good reference to the topic helps.

During my undergraduate course, the Head/Dean of my math department tried very hard to teach me how to rewrite the entire solution as a sequence of logical statements. Sadly, I did not learn all that well. However, as one of the answers above points out, logical statements can be tested (easily enough if you follow the rules) and therefore are something you can work with!

Finally, the one thing that counts above all the advice is that the easiest way to solve something is to solve a bunch of easy problems like yours and then apply the same tricks/rules. If you aren't inventing crazy rules to solve your problem, you are very likely correct. However, if after repeated testing, you think your proof is consistent, that is when you want someone with more experience to take a look. Get a peer-review. :)

I would suggest being resourceful, up to and not including the point of having access to solutions. Even asking help from a peer is a lot more likely to elicit someone giving a constructive hint instead of a less constructive solution.

At this level of sophistication, most textbooks do not have solution manuals written by the author. You may be able to find solutions written by readers, especially for textbooks that have seen long-term and widespread use. For example, a simple search should lead you to several reader-written solution manuals to Rudin's Real and Complex Analysis.

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