R.D Silverman's number theory homework

[davar55]

Originally Posted by CRGreathouse

The Cartesian product of the empty set with anything (in your example, the empty set) is the empty set. All of the members of the empty set are nonempty. (Fun fact: all of the members of the empty set are zebras.)

Why not mention that all empty sets are equal / the same, even though
they may be defined in terms of containing zero number of elements from
different universal sets? BTW what goes on if the universal set U is empty?

to sm88: these are NOT trivial or even easy to fully grasp. But you will,
eventually. Keep on plugging.

[davar55]

Originally Posted by R.D. Silverman

I don't think you want to ask this question here. The answer will involve
some fairly deep combinatorics, (i.e. Polya's Counting Theorem).
[i.e. consider duplicates induced by symmetries, rotations, and reflections]

RDS is correct here. But these comments will help sm88 in the future.

[davar55]

Originally Posted by Mr. P-1

I'm not asking the question you think I am. In particular I did not intend symmetries to be regarded as duplicates.

In fact, my question is just a disguised version of the question asked by CRGreathouse earlier in the thread: if S = {1, 2, 3, 4, 5,}, how many different binary relations are there on S?

to sm88: when you get the definitions, this counting question will be
important to solve.

[davar55]

Counting versus foundations of set theory versus
foundations of algebra. We all went through these,
this is a group collaborative effort to assist one
of our colleagues into advancement in math.

Just wanted to be the first to mention "groups" in this thread.

[Mr. P-1]

Originally Posted by science_man_88

I forgot to reply to CRG's questioning. If and when you're positive I know enough to move on feel free. Next preliminary is equivalence classes.

I don't think you're ready to move on. I think you'd benefit from more playing around with different relations, including some which are not equivalence relations.

Try doing the next case of the transitivity proof of ~ by setting a different pair of variables equal (either x and z, or y and z) and using the part of the proof I did as a model.

[Mr. P-1]

Originally Posted by davar55

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.

That was my assessment of sm88's level earlier in the thread.

Originally Posted by davar55

Why not mention that all empty sets are equal / the same, even though they may be defined in terms of containing zero number of elements from
different universal sets? BTW what goes on if the universal set U is empty?

This would have been much easier for all of us, including sm88, if we'd started with the basics of set theory, and worked our way through unions, intersections, differences, complements, De Morgan's laws, cartesian products, and so on, and only then did relations. As it is, I'm not sure if sm88 even knows what it means for two sets to be equal.

But the relations thing has achieved a certain momentum, so I'm running with it.

[Mr. P-1]

Originally Posted by davar55

In which thread did the illustrious and well respected RDS affectionately
known herein as RDS post these homework problems?

http://www.mersenneforum.org/showthread.php?t=14861

Unfortunately SM88 tried too hard to adhere to the letter of RDS' admonition. As a result this thread has departed markedly from its spirit. If I might paraphrase RDS' advice:

1. Get a maths textbook
2 & 3. Learn some maths from it.
4. Come and talk to us about what you've learnt.

But because he took it literally, he got "an elementary number theory text" which is so way beyond him that he's stuck on the prerequisites.

Texts don't teach prerequisites. They assume you already know the prerequisites. Consequently, SM88 has been learning maths, not from his text, but from us. And we've been willing to teach him, because we're nice people.

It would have been easier, for him and for us, it he'd got a more appropriate text for his level.

[Mr. P-1]

Originally Posted by science_man_88

one partition of a column/row of a chessboard is white squares in the column/row and black squares in the column/row

Forget partitions. You don't know what they are, as we haven't done them yet.

one binary relation that could be drawn is "is the same color as"

Yes, that's a binary relation. What set is this relation on?

if square1 is the same color as square3 and square3 is the same color as square5 then square1 is the same color as square5, this is true and shows transitivity.

Transitivity is a claim about all possible combinations of x, y, and z. You can use a single counterexample to disprove such a claim, but you cannot use a single example to prove it.

Let x, y, and z be any squares on a chess board and let c() be the colour function on x

Then x is the same colour as y implies that c(x) = c(y). (This is not really an implication, more a restatement).

y is the same colour as z implies that c(y) = c(z).

But c(x) = c(y) AND c(y) = c(z) IMPLIES c(x) = c(z)

This is true because equality is a transitive relation. (This is an elementary property of equality; we don't need to prove it.)

And so x is the same colour as z.

Can you prove reflexivity and symmetry, using the above as a model? Hint: equality is more than just transitive. Equality is an equivalence relation.

[Mr. P-1]

Number the rows of a chess board 1 to 8. Likewise number the columns. Let x and y be numbers in the range 1 to 8.

Does the statement "The square at row x, column y is black" define a relation? If so, is it the same relation as the one talked about above?

[R.D. Silverman]

Originally Posted by Mr. P-1

1. Get a maths textbook
2 & 3. Learn some maths from it.
4. Come and talk to us about what you've learnt.

But because he took it literally, he got "an elementary number theory text" which is so way beyond him that he's stuck on the prerequisites.

.

It might be useful if he also gained some understanding of quantifiers
and got some practice in the use of variables.

[xilman]

Originally Posted by R.D. Silverman

It might be useful if he also gained some understanding of quantifiers
and got some practice in the use of variables.

He is now getting a good grounding in the latter; the former will come later, but not too much later.