[QUOTE=Mr. P-1;247892]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.)[/QUOTE]

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.

[QUOTE=xilman;247895]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[/QUOTE]

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 [TEX]\therefore[/TEX] it can't be an equivalence relation.

[QUOTE=science_man_88;247897]but I've likely failed already.[/QUOTE]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

[QUOTE=xilman;247900]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[/QUOTE]

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

[QUOTE=axn;247481]Incorrect. A tuple is an _ordered_ list of elements. Why is that "ordered" part important?[/QUOTE]

[QUOTE=science_man_88;247482]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).[/QUOTE]

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).

[QUOTE=Mr. P-1;247892]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.)[/QUOTE]

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]

[QUOTE=R.D. Silverman;247903]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][/QUOTE]

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.

[QUOTE=science_man_88;247893]1) S[TEX]\times[/TEX]S = {(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 4[TEX]\times[/TEX]4 = 16 elements[/QUOTE]

Correct.

[QUOTE]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 [TEX]\therefore[/TEX] Mother is a sibling of Mother.[/QUOTE]

OK.

[QUOTE]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.[/QUOTE]

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

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

Correct.

[QUOTE]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,[/QUOTE]

Right answer, incomplete justification. What have you missed?

[QUOTE]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[/QUOTE]

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

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

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.

[QUOTE=R.D. Silverman;247903]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][/QUOTE]

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?

[QUOTE=science_man_88;247898]4) because you have defined x!~x this relation can't be reflexive in my mind.[/QUOTE]

Correct.

[QUOTE]from that it can't have all 3 properties needed to be an equivalence relation [TEX]\therefore[/TEX] it can't be an equivalence relation.[/QUOTE]

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

[QUOTE=Mr. P-1;247909]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?

[COLOR="red"]Not to my knowledge.[/COLOR]



Correct.



Right answer, incomplete justification. What have you missed?

[COLOR="red"]The words symmetric ?[/COLOR]


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

[COLOR="Red"]if you want one that doesn't repeat I see no other as the relations in the list I gave number just 2.[/COLOR]


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.[/QUOTE]

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.