Do I have this definition right? (and a suprise bonus question on sequences)

[quasar987]

Definition: Suppose T is an index set and for each t in T, X_t is a non-void set. Then the product $\Pi_{t \in T}X_t$ is the collection of all "sequences" $\{x_t\}_{t \in T} = \{x_t\}$ where $x_t \in X_t$.


Does this mean that $\Pi_{t\inT}X_t$ is the set containing all possible sequences defined by: "i-th element is a member of the set X_i"?

And by the way, if sequences are sets of ordered elements, why aren't they noted using parentheses instead of braces? Afterall, isn't an n-tuple (x_1,...,x_n) just a set whose elements are ordered, i.e. a sequence?

 

[quasar987]

I've stumbled on this other definition of sequence, also in mathworld, that goes

"A family with index set $\mathbb{N}$ is called a sequence."

This one makes no reference to the fact that the elements are ordered though.

 

[matt grime]

There is no reason, for a general index set, why it should be ordered, or even countable.

the product is a set with a certain set of properties. You should think of it as being something that has exactly one copy of each set X_t for t in T inside it, and maps, called projections from the product to each of the factors X_t.

If T were the natural numbers then you could think of it as sequences, but it is not necessary and is unhelpful for the general case. In the general case the index set is not ordered so it is not possible to think of it as an ordered T-tuple (what would that mean if T were the real numbers say?)

 

[quasar987]

I see. Thanks matt.

 

[mathwonk]

elements of a product are functions on the index set, with values in the factor sets.

 

[HallsofIvy]

I was just looking over an elementary analysis text where the author notes that he is using (a_n) to represent a sequence rather than the more standard {a_n} because he wants to emphasize that a sequence is not just a set but an ordered set. This seems to me to be missing the point that the chief questions about a sequence: whether or not a sequence converges and, if so, to what it converges, are independent of the order.

I have always thought that it was enough to define a sequence to be a countable set- so that there was some order for the set- but it didn't matter which of the infinite number of possible orderings you chose.

 

[mathwonk]

halls, you seem to be right that the limit definition of convergence of a sequence is independent of order, and that is a nice remark, but the most elementary criterion for convergence of a sequence, namely monotone and bounded, is very dependent on order, such as for the elementary definition of the limit of an infinite decimal.

the meaning of the word "sequential" also seems to require order, as opposed to merely countable cardinality.

also the phenomenon of conditional convergence requires ordering for the terms of a series, which is, in the usual sense, merely a sequence with plus signs inserted.

sequences with orderings are also crucial for defining countable ordinals. Even in defining limits the ordering gives a way to mark off finite from cofinite sets of elements of the sequence. it also allows a measure of speed of convergence by comparing n to epsilon.

as to proving convergence, instead of mjerely defining it, that may well use ordering also in most actual cases.


so perhaps it is for the other uses of ordered sequences, besides the mere definition of convergence, that the ordering is considered crucial.

nice remark though. that would make a good exercise for analysis students.

the author of that analysis text however seems to miss the point of ordering. it is not embodied in the choice of bracket notation, but in the functional relation implied by the subscript. i.e. "a sub n" is the term that follows "a sub (n-1)" no matter what kind of brackets surround it.

i.e. to dispense with the ordering, he would have to use an unordered index set. i.e. the "unordered set" {a2 , a6, a1, a5, a3, a4}, still has a unique ordering given by the ordering of the indices 1,2,3,4,5,6.

a sequence is merely a function defined on the positive integers, i.e. a set of ordered pairs whose first elements are the positive integers, and each pair is still ordered, no matter what sequence the ordered pairs are presented in. hence n is still associated with a unique element an.