When I said R is the closure of Q and you called it instead the completion,
weren't we both right?

[QUOTE=davar55;250201]When I said R is the closure of Q and you called it instead the completion,
weren't we both right?[/QUOTE]

The closure of [b]Q[/b] isn't defined absolutely. You need to specify a superspace to take the closure in.

[QUOTE=science_man_88;249252]are there any other terms I should define ?[/QUOTE]

subset.

[QUOTE=Mr. P-1;250307]subset.[/QUOTE]

a set within a set or a collection of sets where each set in the collection is 1 element from the larger set.

[QUOTE=Mr. P-1;249147]An intersection between two sets is [i]the set[/i] whose elements are just the elements that the two sets have in common.

1. We haven't talked about it yet, but how do you think you could define the intersection of a collection of sets?

The union of two sets is [i]the set[/i] whose elements are members of at least one of the two sets.

2. Try to define the union of a collection of sets.

A Cartesian product A x B of two sets is the set of all ordered pairs (a, b) with a[TEX]\in[/TEX]A and b[TEX]\in[/TEX]B.

3. (Hard) How would you define a Cartesian product on an n-tuple of sets?[/QUOTE]

SM88, do you think you could have a go at these?

[QUOTE=science_man_88;250309]a set within a set or a collection of sets where each set in the collection is 1 element from the larger set.[/QUOTE]

Too vague. Try completing the following:

A subset of a set S is a set whose members are...

[QUOTE]1. We haven't talked about it yet, but how do you think you could define the intersection of a collection of sets?

The union of two sets is the set whose elements are members of at least one of the two sets.

2. Try to define the union of a collection of sets.

A Cartesian product A x B of two sets is the set of all ordered pairs (a, b) with aA and bB.

3. (Hard) How would you define a Cartesian product on an n-tuple of sets?[/QUOTE]

1) intersection of all the sets within the collection ?

2) the union on all sets within the collection ?

3) a n-tuple of sets comes from picking a element from n not necessarily distinct collections no ? I'm not completely sure sounds like a Cartesian product within a Cartesian product to me.

[QUOTE=science_man_88;250313]1) intersection of all the sets within the collection ?

2) the union on all sets within the collection ?[/QUOTE]

Yes. But...

Your previous definitions of intersection and union were for just two sets. You haven't defined these concepts for more (or for that matter, for less) than two sets.

Try completing these sentences.

The intersection of two sets, A and B, is the set whose elements are...

The union of two sets, A and B, is the set whose elements are...

The intersection of a collection of sets, C, is the set whose elements are...

The union of a collection of sets, C, is the set whose elements are...

[QUOTE]3) a n-tuple of sets comes from picking a element from n not necessarily distinct collections no ? I'm not completely sure sounds like a Cartesian product within a Cartesian product to me.[/QUOTE]

I did say it was hard.

The Cartesian product of an n-tuple of sets (A[sub]1[/sub], A[sub]2[/sub], ..., A[sub]n[/sub]) is the set of all n-tuples (a[sub]1[/sub], a[sub]2[/sub], ..., a[sub]n[/sub]) with a[sub]k[/sub][TEX]\in[/TEX]A[sub]k[/sub]. for 1[TEX]\le[/TEX]k[TEX]\le[/TEX]n.

[QUOTE=Mr. P-1;250334]Yes. But...

Your previous definitions of intersection and union were for just two sets. You haven't defined these concepts for more (or for that matter, for less) than two sets.

Try completing these sentences.

The intersection of two sets, A and B, is the set whose elements are...

[COLOR="Red"]exactly those elements contained in both A and B, casting out repeats ( if it's in A and B twice)[/COLOR]

The union of two sets, A and B, is the set whose elements are...

[COLOR="red"]exactly all elements of A and B shown once, casting out repeats( if it's in A and B twice)[/COLOR]

The intersection of a collection of sets, C, is the set whose elements are...
[COLOR="red"]in every set within C, casting out repeats( if it's all sets in C more than once)[/COLOR]


The union of a collection of sets, C, is the set whose elements are...

[COLOR="Red"]exactly all elements of the sets within C shown once, casting out repeats( if it's in A and B twice)[/COLOR]




I did say it was hard.

The Cartesian product of an n-tuple of sets (A[sub]1[/sub], A[sub]2[/sub], ..., A[sub]n[/sub]) is the set of all n-tuples (a[sub]1[/sub], a[sub]2[/sub], ..., a[sub]n[/sub]) with a[sub]k[/sub][TEX]\in[/TEX]A[sub]k[/sub]. for 1[TEX]\le[/TEX]k[TEX]\le[/TEX]n.[/QUOTE]


is this what you wanted ?

[QUOTE=Mr. P-1;250278]The closure of [B]Q[/B] isn't defined absolutely. You need to specify a superspace to take the closure in.[/QUOTE]

I see. But by rationals I mean the rational reals, as usual.
So the superspace is R.

Hence, within R, the closure of Q is the completion of Q, which is R.

[QUOTE=davar55;248940]Infinity ... is not a "Real Number" as in the closure of the rational numbers.[/QUOTE]

[QUOTE=davar55;250365]I see. But by rationals I mean the rational reals, as usual.
So the superspace is R.

Hence, within R, the closure of Q is the completion of Q, which is R.[/QUOTE]

Then I fail to see the relevance of referring the closure of [b]Q[/b] in your original comment. That "infinity" is not an element of [b]R[/b] is not a consequence of [b]R[/b] being the closure of [b]Q[/b] in [b]R[/b].

On the other hand, that "infinity" is not an element of [b]R[/b] [i]is[/i] a consequence of [b]R[/b] being the completion of [b]Q[/b] in so far as the latter statement is the [i]definition[/i] of [b]R[/b].