[QUOTE=CRGreathouse;247824]

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

doh! missed this part.

if S is 2 generations of a family, if a~b when ~ is " is a child of" then you can set up an equivalence relation of

[B]a[/B] is a sibling of [B]a[/B]
[B]a[/B] is a child of [B]b[/B] implies [B]b[/B] is a parent of [B]a[/B]

and finally:

[B]a[/B] is a child of [B]b[/B] and [B]b[/B] is a parent of [B]c[/B] implies [B]a[/B] is a sibling of [B]c[/B]. But I may be wrong.

[QUOTE=science_man_88;247831]doh! missed this part.

if S is 2 generations of a family, if a~b when ~ is " is a child of" then you can set up an equivalence relation of

[B]a[/B] is a sibling of [B]a[/B]
[B]a[/B] is a child of [B]b[/B] implies [B]b[/B] is a parent of [B]a[/B]

[/QUOTE]

You are "wrong" in the sense that you have discussed [B]three[/B] different relations (parent of, child of, and sibling of).
Try to confine yourself to the request for [B]a[/B] (single) relation and discuss properties (such as reflexivity and transitivity) about that relation.

[QUOTE=Wacky;247832]You are "wrong" in the sense that you have discussed [B]three[/B] different relations (parent of, child of, and sibling of).
Try to confine yourself to the request for [B]a[/B] (single) relation and discuss properties (such as reflexivity and transitivity) about that relation.[/QUOTE]

so in other words relations like is a sibling of work better because

x is a sibling of x
x is a sibling of y implies y is a sibling of x

and x is a sibling of y and y is a sibling of z implies x is a sibling of z of course this violates my definition of S.

may go to bed after the next show on tv.

[QUOTE=science_man_88;247833]and x is a sibling of y and y is a sibling of z implies x is a sibling of z of course this violates my definition of S.[/QUOTE]

Why?

[QUOTE=science_man_88;247826]okay what about the Cartesian product of a set S such that S is the empty set, then technically wouldn't you only match up nothing with nothing, but I guess it's not the same.[/QUOTE]

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.)

[QUOTE=CRGreathouse;247838]Why?[/QUOTE]

because I said 2 generation but I'm only involving one ? Nm I thought I would have to use both generations I'm not thinking it's past when I usually goto bed. The subset R doesn't have to involve both unless we have one set for each individual generation.

[QUOTE]A more complicated exercise: implement the same output in BASIC, Perl, C, C++, C#, Java, Javascript or any other programming language of your choice and comfort....[/QUOTE]Our head hurts real bad from reading this thread, but we are always game for (extremely simple) programming exercises.

[CODE]? for [ a 1 12 ] [ type form :a 4 0 for [ b 2 12 ] [ type form product :a :b 4 0 ] ( print ) ]
1 2 3 4 5 6 7 8 9 10 11 12
2 4 6 8 10 12 14 16 18 20 22 24
3 6 9 12 15 18 21 24 27 30 33 36
4 8 12 16 20 24 28 32 36 40 44 48
5 10 15 20 25 30 35 40 45 50 55 60
6 12 18 24 30 36 42 48 54 60 66 72
7 14 21 28 35 42 49 56 63 70 77 84
8 16 24 32 40 48 56 64 72 80 88 96
9 18 27 36 45 54 63 72 81 90 99 108
10 20 30 40 50 60 70 80 90 100 110 120
11 22 33 44 55 66 77 88 99 110 121 132
12 24 36 48 60 72 84 96 108 120 132 144[/CODE]

[QUOTE=science_man_88;247833]so in other words relations like is a sibling of work better because

x is a sibling of x
x is a sibling of y implies y is a sibling of x

and x is a sibling of y and y is a sibling of z implies x is a sibling of z of course this violates my definition of S.

may go to bed after the next show on tv.[/QUOTE]

Let's make this example more concrete. Let S={Mother, Father, Son, Daughter}, where Son and Daughter are the children of Mother and Father (who are, one hopes, not brother and sister to each other.)

1. Write out in full the Cartesian Product S x S. How many elements does this Cartesian Product have?
2. Is Mother a sibling of Mother?
3. Write out in full the relation "is a sibling of" on S
4. Is this relation a: reflexive, b: symmetric, c transitive, d: an equivalence relation? Justify your answer in each case

Note that there is no right or wrong answer to question 2. This is just to get you to define more precisely what "is a sibling of" actually means. Your answers to 3 and 4 will, however depend upon your answer to question 2.

[QUOTE=CRGreathouse;247801]Challenge question: If S is {1, 2, 3, 4, 5} and S x S is {(1, 1), (1, 2), ..., (4, 5), (5, 5)} there are 5 elements in S and 25 elements in S x S. How many binary relations are there on S?[/QUOTE]

[QUOTE=science_man_88;247805]well if each 2-tuple has one relation on it then it would have to be 25. If you say less than,less than or equal to, equal to, greater than or equal to, and greater than , then it's at least 5 but my best guess is 25.[/QUOTE]

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=Mr. P-1;247888]Let's make this example more concrete. Let S={Mother, Father, Son, Daughter}, where Son and Daughter are the children of Mother and Father (who are, one hopes, not brother and sister to each other.)

1. Write out in full the Cartesian Product S x S. How many elements does this Cartesian Product have?
2. Is Mother a sibling of Mother?
3. Write out in full the relation "is a sibling of" on S
4. Is this relation a: reflexive, b: symmetric, c transitive, d: an equivalence relation? Justify your answer in each case

Note that there is no right or wrong answer to question 2. This is just to get you to define more precisely what "is a sibling of" actually means. Your answers to 3 and 4 will, however depend upon your answer to question 2.[/QUOTE]

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

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.

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.

4) As my definition allows a single person to be a sibling of themselves the reflexive is proven,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, 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 proving it transitive , finally by virtue of having the other 3 properties this is clearly an equivalence relation.

[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

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.

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.

4) As my definition allows a single person to be a sibling of themselves the reflexive is proven,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, 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 proving it transitive , finally by virtue of having the other 3 properties this is clearly an equivalence relation.[/QUOTE]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