**Over Sets**

If *n* is a natural number and *A* is an arbitrary set, the expression *A**n* is often used to denote the set of ordered *n*-tuples of elements of *A*. This is equivalent to letting *A**n* denote the set of functions from the set {0, 1, 2, …, *n*−1} to the set *A*; the *n*-tuple (*a*_{0}, *a*_{1}, *a*_{2}, …, a_{n−1}) represents the function that sends *i* to *a*_{i}.

For an infinite cardinal number κ and a set *A*, the notation *A*κ is also used to denote the set of all functions from a set of size κ to *A*. This is sometimes written κ*A* to distinguish it from cardinal exponentiation, defined below.

This generalized exponential can also be defined for operations on sets or for sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of

where each *V*_{i} is a vector space.

Then if *V*_{i} = *V* for each *i*, the resulting direct sum can be written in exponential notation as *V*⊕**N**, or simply *V***N** with the understanding that the direct sum is the default. We can again replace the set **N** with a cardinal number *n* to get *V**n*, although without choosing a specific standard set with cardinality *n*, this is defined only up to isomorphism. Taking *V* to be the field **R** of real numbers (thought of as a vector space over itself) and *n* to be some natural number, we get the vector space that is most commonly studied in linear algebra, the Euclidean space **R***n*.

If the base of the exponentiation operation is a set, the exponentiation operation is the Cartesian product unless otherwise stated. Since multiple Cartesian products produce an *n*-tuple, which can be represented by a function on a set of appropriate cardinality, *S**N* becomes simply the set of all functions from *N* to *S* in this case:

This fits in with the exponentiation of cardinal numbers, in the sense that |*S**N*| = |*S*||*N*|, where |*X*| is the cardinality of *X*. When "2" is defined as {0, 1}, we have |2*X*| = 2|*X*|, where 2*X*, usually denoted by **P**(*X*), is the power set of *X*; each subset *Y* of *X* corresponds uniquely to a function on *X* taking the value 1 for *x* ∈ *Y* and 0 for *x* ∉ *Y*.

Read more about this topic: Complex Numbers Exponential, Generalizations

### Other articles related to "set, sets":

1) Transaction

**Set**(997) will be replaced by Transaction

**Set**(999) "acknowledgement report" ... have been removed from existing Transaction

**Sets**... segments have been added to existing Transaction

**Sets**allowing greater tracking and reporting of cost and patient encounters ...

... Box

**sets**can contain collections of items other than music, video, or literary media ... Some examples include a box

**set**of basic carpentry tools, a barbecue cooking utensil box

**set**, and a box

**set**of loose-leaf tea varieties ... Box

**sets**generally offer a number of items normally sold separately as one

**set**...

... In the case of music, contemporary box

**sets**are usually made up of four or more discs boxes, covering a broad range of boxes of the music of a given artist or genre ... collections of their boxes of music released as box

**sets**... Some box

**sets**collect together previously released boxes of singles or albums by a music artist, and often collect the complete discography of an artist such as Pink Floyd's Oh, by the ...

... Given a

**set**X and an indexed family (Yi)i∈I of topological spaces with functions the initial topology τ on is the coarsest topology on X such that each is continuous ... topology may be described as the topology generated by

**sets**of the form, where is an open

**set**in ... The

**sets**are often called cylinder

**sets**...

... of maps {fi X → Yi} separates points from closed

**sets**in X if for all closed

**sets**A in X and all x not in A, there exists some i such that where cl denoting the ... continuous maps {fi X → Yi} separates points from closed

**sets**if and only if the cylinder

**sets**, for U open in Yi, form a base for the topology on X ... It follows that whenever {fi} separates points from closed

**sets**, the space X has the initial topology induced by the maps {fi} ...

### Famous quotes containing the word sets:

“Willing *sets* you free: that is the true doctrine of will and freedom—thus Zarathustra instructs you.”

—Friedrich Nietzsche (1844–1900)

“It is mediocrity which makes laws and *sets* mantraps and spring-guns in the realm of free song, saying thus far shalt thou go and no further.”

—James Russell Lowell (1819–91)