Question

How to read this notation, and make sense of it:

AxB := { {x, {x,y}} : x in A, y in B }

A cross B is defined to be the set of ..... i got no idea how to make sense of the {x,[x,y]} part of it

Answer 1

this is somehow spose to define the ordered pair (x,y) but i just cant make heads or tails of it .... topology by the way

Answer 2

to me, x in A and {x,y} in B

Answer 3

might help to see it in the book ...

Answer 4

http://en.wikipedia.org/wiki/Cartesian_product#Most_common_implementation_.28set_theory.29

Answer 5

It does say that \[\Large \{x,\{x,y\}\}\] is just a fancy way of writing \[\Large (x,y)\]
Since, later on on that page, it redefines it, sort of...

Answer 6

yeah, i did read past it lol ... im just wondering how to comprehend what the first set of subsets? ... reads out as.

Answer 7

the set with elements x and {x,y} such that x in ... y in ...

Answer 8

As per the wikipedia link (for all it's worth)
that {x , {x,y}} is supposed to be a set of sets...though, I still don't see what use that serves... yeah... I'm lost XD

Answer 9

the karatowski definition eh

Answer 10

maybe just look at it as organasing the cartesian product set by elements with comun 1º element and different 2º element. So you would have as many groups as different 1º elements are there

Answer 11

just leaves me wondering what 1 and 2 degree elements mean :/

Answer 12

I think this is to make consistent with set theories ... such that (x,y) is itself a set.

Answer 13

I suspect Kuratowski was drunk :>
Maybe this way, you are spared from having to define a 'first' coordinate and a 'second' coordinate for your ordered pair, since your first coordinate is the only element that appears in all the sets in {{x} , {x,y}} ...

Answer 14

let A = {1,2,3}
let B = {a,b}

AxB = {(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}

and this is the power set of (the power set of(AuB))

Answer 15

Don't you mean a subset of the powerset of A U B ? :3

Answer 16

:)

Answer 17

AuB = {1,2,3,a,b} ... no, a subset of the power set of the powerset of lol

Answer 18

Oh yeah >.>
Blasted nests...

Answer 19

under that def ... i think it's not subset of power set of power ...
on this Kru...'s def ... i think it will be subset of ...

Answer 20

the power set of AuB is: well, it has 2^5 elements ....

does that mean that there are 2^(2^5) elements of AxB ?

Answer 21

No... since AxB isn't exactly equal to the powerset of the powerset of blablabla
It's just a subset, right?

Answer 22

no ... there is 2*3 elements of AxB so AxB is subset of power set of ...

Answer 23

then why did i get 6 elements for AxB?

Answer 24

Simply because there are? :3

Answer 25

because there are six elements of AxB

Answer 26

...six elements, I mean.

Answer 27

...what eX said :D

Answer 28

lol, im sure you typed 2^3 first

Answer 29

Wasn't that 2*3? D:

Answer 30

there is something like every object is a set itself in some set theories like ZF set theory or naive set theory. IDK i haven't studied.
(a,b) is simply a tuple ... not a set itself. also \( (a,b) \) is not subset of that power set.

but under this def (a, b) = {a, {a,b}} it becomes subset of that power whatever set and consistent with those stupid set theories. .... I think but not sure.

Answer 31

this is what we get for letting the germans play with numbers ...

Answer 32

{a, {a,b}} is not element of set ... but a set itself.

Answer 33

LOL

Answer 34

but Kuratowski was Polish? :/

Answer 35

Poland was occupied wasnt it?

Answer 36

Oh great.. history... well, his part of Poland was Russian-occupied, not Prussian >:)

Answer 37

I thought he was russian ... many russian names ends with 'i' or 'y'

Answer 38

:) ive got to get back to data entry before i cant pay the rent ... thanx yall

Answer 39

To avoid ambiguity, Russian '-ski' names use '-sky' instead, so that '-ski' usually indicates a Polish name... and should we really go off-topic here? (on a mod's question, no less XD)

Answer 40

good luck (y)

Answer 41

edit *{a, {a,b}} is not just element of set ... but a set itself.