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 > <stephe...@charter.net>wrote: > >> >> 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 >> initio<http://en.wikipedia.org/wiki/Ab_initio>. >> >> >> >> > > 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 > sense. > > Jesse > -- > > 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." -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.
