On Wed, May 30, 2012 at 2:02 AM, Stephen P. King <stephe...@charter.net>wrote:
> On 5/29/2012 11:46 PM, Jesse Mazer wrote:
> On Tue, May 29, 2012 at 10:49 PM, Stephen P. King
>> Hi Jesse,
>> Would it be correct to think of "arbitrary" as used here as meaning "
>> some y subset Y identified by some function i or mapping j that is not a
>> subset (or faithfully represented) in X, yet x => y : x /subset X"? The
>> "choice" of a basis of a linear space comes to mind. The idea is that one
>> it is not necessary to specify the method of identification ab
> I can't really tell what you're asking here. As I said, "an arbitrary
> member of set Y will have property X" just means "every member of set Y has
> property X", nothing more complicated. For example, Y might be the set of
> all triangles in Euclidean geometry, and X might be the property of having
> all the inner angles add up to 180 degrees. It would be easier to
> understand your question if you similarly supplied some simple of what Y,
> y, j, X, and x could stand for, such that your description above would make
> Hi Jesse,
> You previously wrote: "The notion of "choosing" isn't actually
> important--if a proof says something like "pick an arbitrary member of the
> set X, and you will find it obeys Y", this is equivalent to the statement
> "every member of the set *X obeys Y*"." and not " "an arbitrary member of
> set Y will have property X" just means "every member of set Y has property
> X" ", a small but possibly important difference.
> Are you assuming a commutative relation for Y and X? Details...
Sorry, I was speaking informally, so I wasn't being too careful about
keeping my use of the symbols X and Y consistent from one post to another.
In the sentence "an arbitrary member of set Y will have property X" I was
using "Y" to refer to a set and "X" to refer to a property, while in the
sentence "pick an arbitrary member of the set X, and you will find it obeys
Y" it was X that referred to a set, and Y that referred to a property. I'll
try to stick to the second usage from now on to be consistent. Also, if I
was worrying more about notation it would be more standard to use Y(x) to
refer to the notion that some mathematical object x has a property Y, and
then if X refers to a set, I could write ∀x (x ∈ X -> Y(x) ), which means
"for all objects x in our domain of discourse, if x is a member of the set
X, this implies that x has property Y."
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