R.D Silverman's number theory homework

[davar55]

Originally Posted by CRGreathouse

My example doesn't have anything to do with specific variables (x, y, or z). It just says that any two numbers are related exactly when their difference is an integer. If you have the variables z and z, then z ~ z exactly when z - z is an integer. If you have the variables c and d, then c ~ d exactly when c - d is an integer. [Edit: Chris Card explains this above.]

So is this an equivalence relation or not? If not, which property/properties fails/fail?

Second exercise: Let m and n be integers. Then m ~ n if and only if 1 + 1 = 2. Is this an equivalence relation?

Third exercise: Let x and y be real numbers. Then x ~ y if and only if xy = 2. Is this an equivalence relation?

First Exercise from CRG: checked all three properties by CC, hence yes
it is an equivalence relation.

Second Exercise: Hint from me: since 1 + 1 = 2 is always true, then
all m are related to all n. From that, now check the three properties.

Third Exercise: Hint from me: if xy = 2, then given x, what are the y
that are equivalent to x? Then check the three properties.

(note to CRG: good ones. Later modular equality. Obviously.)

[davar55]

Originally Posted by R.D. Silverman

Yep. It is an ordering relation, assuming the standard topology on R.

If sm88 is having trouble with the current problems, perhaps he might
find it easier to determine if simple equality (on R, or Z, or Q) is
an equivalence relation.

RDS seems to assume a more advanced student.
Some students struggling with the definitions of sets, relations,
equivalences, and partitions are only at the junior high level.

[davar55]

Originally Posted by CRGreathouse

OK, I'll do it for the 'birthday' example, then you try to do it for mine.
Transitivity means: if a ~ b and b ~ c then a ~ c.
Our relation is: x ~ y means "x has the same birthday as y".

So writing this out in full for this case, it becomes: if a has the same birthday as b, and b has the same birthday as c, then a has the same birthday as c.

(I've used colors to show which things I substituted for x and which things I substituted for y in the definition of the relation.)

Then I examine the statement and determine (from what I know of birthdays) that it is true. Thus the "has the same birthday as" relation is transitive. The other two properties work out as well, so it's an equivalence relation.

So do what I just did, but for x ~ y means "x - y is an integer" in the definition of transitivity.



See above. At any point you're just working with two things: what's on the left of the ~ and what's on the right of the ~.

In the birthday example:

REFLEXIVITY: Does x always have the same birthday as x? When you
can say to yourself "Of course she does" then you'll understand reflexivity.

SYMMETRY: If x had the same birthday as y, does that mean y has the same one as x? For birthdays, you'll come to the same "Of course".
Then you'll understand symmetry of relations.

TRANSITIVITY: If x has ... as y and y has ... as z does x have ... as z?
Get that "Aha!" and you'll grasp transitivity.

Then try the other proposed relations.

[davar55]

Originally Posted by science_man_88

well a-b+b-c = a+(-b)+(+b)-c = a+0-c = a-c so it looks realizable, according to my earlier post it was symmetric and reflexive. so I guess it has to be yes. though I still have no idea what I'm doing to be honest.

Hey I remember feeling the same way once in a different
math learning context. The meaning came well AFTER the
formal instruction.

[davar55]

Originally Posted by CRGreathouse

Agreed. We have different approaches: you want to teach the underlying formal mechanics and I want to get him accustomed to actually using relations. Since he'll need to know the mechanics (Cartesian products, tuples, etc.) anyway, I don't mind your approach -- but I think it's much harder than just giving an informal definition (like xilman's relation machine) and working through it.

I *don't* think that it's important to understand relations-as-sets, at this point anyway. I would prefer to teach someone to add and only later teach them the Peano postulates.

Yes, and my hope is that once he has developed some intuition by working with some concrete relations he'll be able to move on to the abstractions as needed. But frankly I don't think they're critical here; there are many other gaps I would fill first before moving on to that.

Having said that, if your approach works here, great!

As probably we all did.

[davar55]

Originally Posted by science_man_88

finally a no I was looking forward to lol.

That "finally" is an "aha!", isn't it?

[davar55]

Originally Posted by science_man_88

had to resize it to about 30% original size ( after selecting resize to fit) per side to work.

I see you're analyzing the 8x8 board for chess positions/moves.

I remember doing a similar analysis of the 8x8 board when
I wrote an Othello-playing program. Looked similar.

[davar55]

Originally Posted by science_man_88

actually it still is as you didn't come to 312 either. I won't bother posting what I said before deletion because before starting this thread I asked if prime 95 felt like deleting my account as I felt worthless now either way I have no reason to care. also I can make mine work I just didn't think it through and I was doing it another way. I don't care if I'm wrong anymore I gave up being right when I came on why go off being accurate.

sm88 -- if no one else understood this post, I guarantee you i did.

[davar55]

Originally Posted by jyb

First of all, Cartesian products are not a subject of number theory. They are a very basic part of set theory, which underlies all of mathematics. By learning about them, you haven't even started to get to number theory yet. That's why they're in the "preliminaries" section.

Second, relating your counting of chess moves to Cartesian products by pointing out that you can label the squares using the product of your two sets would be like me noticing that the street outside is named 68th and claiming that a list of street names in my city could be helpful for understanding mathematics ("I relate this to my example of a street that has a number in its name").

Why don't you follow Paul's advice and get back to the subject that started this thread? I think it has been adequately demonstrated that there are many people who would be happy to help you learn. But that good will will quickly evaporate if you can't stay on topic, recognize your own limitations, make decent attempts at the suggested problems, and stay on topic.

Since it was a subject that was causing you trouble, have you given any further thought to relations? Do you feel that you can come up with some examples of your own? Can you now understand the formal definition?

Stop telling sm88 to "recognize his own limitations".
It's up to the teachers to focus the student.

[davar55]

Originally Posted by science_man_88

If you mean the word element which would imply a set last i checked. thought you didn't want people giving me hints lol, though you didn't give me it outright.

Hints are necessary, but shouldn't give away the farm.

[davar55]

Originally Posted by CRGreathouse

It's actually A and A, but no big deal. (The Cartesian product FALSE is empty, but the *elements* of a Cartesian product are never empty -- they're ordered pairs, and no matter what you choose for a and b, (a, b) is not the same as {}.)

So, your turn: Pick a set (call it S) and make a relation on S.

Just want to let sm88 know that this kind of comment is from an advanced
viewpoint that already understands the relationships among set theory,
symbolic logic, and even computer programming theory.

It might be a good idea for a good but not advanced student to
get sidetracked on the side ...