R.D Silverman's number theory homework

[science_man_88]

Originally Posted by Mr. P-1

CRGreathouse defined a set S, and asked about binary relations on S. He did not ask about relations on 2-tuples, (pairs from now on).

Here's another question: You have a chessboard, except that it is small, only five squares to a side. You also have 25 white and 25 black counters. How many different ways could you place exactly one counter on each square? (Obviously you will have 25 counters left over.)

Well all the subsets of S in this case should be of the form (X one color +(25-X) of the other) switching between colors I get 50 unique subsets. within each subset it then moves from amount of a color to placement, in the full color subsets there's only one unique placement of color for a total of 2, in the ones with (25-X) of the other color there are at least 25-X combinations , I'm pretty sure I'd be using factorials for placement, but I've likely failed already.

[science_man_88]

Originally Posted by xilman

Your analysis is correct, given your definition of "sibling". Good. You're making real progress.

Now try again with a slightly different definition of "sibling". Define a sibling of a person to be either a brother of that person or a sister of that person but excluding the person him/herself.


With this definition, is it still an equivalence relation? If not, why not? If so, prove it as you did with your definition.

Paul

1) stays the same,

2) basically your stating that x!~x

3) the full list would them come to: son is a sibling of daughter, daughter is a sibling of son.

4) because you have defined x!~x this relation can't be reflexive in my mind. from that it can't have all 3 properties needed to be an equivalence relation it can't be an equivalence relation.

[xilman]

Originally Posted by science_man_88

but I've likely failed already.

Your continuous display of pessimism and self-denigration is getting boring. Please try to stop posting it and to stick to the subject of the discussion.

Paul

[science_man_88]

Originally Posted by xilman

Your continuous display of pessimism and self-denigration is getting boring. Please try to stop posting it and to stick to the subject of the discussion.

Paul

Sorry Paul I didn't see an easy way forward ( I know not all paths are easily proven, so I must stop doing this).

[Mr. P-1]

Originally Posted by axn

Incorrect. A tuple is an _ordered_ list of elements. Why is that "ordered" part important?

Originally Posted by science_man_88

Because it allows people to know things in a specific order by the sounds of it. Without order there is chaos ( wonder if that's why politics doesn't work).

You have the wrong meaning of "ordered" in your head. In this context, "ordered" does not mean orderly (vs. disorderly, chaotic). It means "in a particular order" (vs. "in no particular order".)

Ordinary sets are unordered. The set {Mother, Father} is the same as the set {Father, Mother}. Pairs are ordered. (Mother, Father) is different from (Father, Mother).

[R.D. Silverman]

Originally Posted by Mr. P-1

CRGreathouse defined a set S, and asked about binary relations on S. He did not ask about relations on 2-tuples, (pairs from now on).

Here's another question: You have a chessboard, except that it is small, only five squares to a side. You also have 25 white and 25 black counters. How many different ways could you place exactly one counter on each square? (Obviously you will have 25 counters left over.)

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]

[science_man_88]

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]

Well I've come across something similar in number freak but it talks about Burnside's lemma. I just haven't wrapped my brain around many questions like this.

[Mr. P-1]

Originally Posted by science_man_88

1) SS = {(mother,mother),(mother,father),(mother,son),(mother,daughter),(father,mother),(father,father),(father,son),(father ,daughter),(son,mother),(son,father),(son,son),(son,daughter),(daughter,mother),(daughter,father),(daughter,son),(daughter,daughter)}
this cartesian product has 44 = 16 elements

Correct.

2) in my definition of sibling it can roughly be defined as having at least one parent in common(at home or biologically), as a person themselves has all parents in common with themselves this fits my definition of being a sibling Mother is a sibling of Mother.

OK.

3) I realize you're pointing out that my relation only works out if I define the family to have at least three(unless they overlap) kids in common with a parent. The full relation for "is a sibling of" would be: mother is a sibling of mother,father is a sibling of father, son is a sibling of son, son is a sibling of daughter, daughter is a sibling of son, and daughter is a sibling of daughter.

I wasn't pointing out anything, and your relation works fine. Your answer is correct. Can you write this out in set-theoretical notation?

4) As my definition allows a single person to be a sibling of themselves the reflexive is proven,

Correct.

Also as in this example the son and daughter would share a parent and hence be siblings which implies that they are siblings so yes daughter is a sibling of son implies son is a sibling of daughter,

Right answer, incomplete justification. What have you missed?

if you allow for overlap you can easily do the last property with just one child but son is a sibling of daughter and daughter is a sibling of son can imply again that son is a sibling of daughter

Incorrect, though on the right lines. What is the correct implication?

finally by virtue of having the other 3 properties this is clearly an equivalence relation.

Correct. One thing to bear in mind, though, is that you have shown this only for this particular family (I.e., this particular Set). "is a sibling of" applied to a different set might have different properties, even with the same "at least one parent in common" definition.

[Mr. P-1]

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]

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?

[Mr. P-1]

Originally Posted by science_man_88

4) because you have defined x!~x this relation can't be reflexive in my mind.

Correct.

from that it can't have all 3 properties needed to be an equivalence relation it can't be an equivalence relation.

Correct, but what about the other two properties. Do they hold?

[science_man_88]

Originally Posted by Mr. P-1

Correct.



OK.



I wasn't pointing out anything, and your relation works fine. Your answer is correct. Can you write this out in set-theoretical notation?

Not to my knowledge.



Correct.



Right answer, incomplete justification. What have you missed?

The words symmetric ?


Incorrect, though on the right lines. What is the correct implication?

if you want one that doesn't repeat I see no other as the relations in the list I gave number just 2.


Correct. One thing to bear in mind, though, is that you have shown this only for this particular family (I.e., this particular Set). "is a sibling of" applied to a different set might have different properties, even with the same "at least one parent in common" definition.

I know you want more complete understanding, But I don't see how I can get the relation not to repeat, the only thing I can figure for the missing in the symmetric proof is the word symmetry. I'm unsure what you mean by set- theoretical notation.