Domain of images of sets under mappings
Dom::ImageSet()
Dom::ImageSet(f
,x
,S
)
Dom::ImageSet(f
,[x1, …]
,[S1, …]
)
Dom::ImageSet
is the domain of all sets of
complex numbers that can be written as the set of all values taken
on by some mapping, i.e., sets of the form
for
some function f and
some sets S_{1},
…, S_{n}.
Image sets are mainly used by solve
to express sets like
.
Dom::ImageSet
belongs to the category Cat::Set
—arithmetical
and settheoretic operations are inherited from there.
Dom::ImageSet(f, x, S)
represents the set
of all values that can be obtained by substituting some element of S
for x
in
the expression f
.
Dom::ImageSet(f, [x1, ...], [S1, ...])
represents
the set of all values that can be obtained by substituting, for each i,
the identifier x_{i} by
some element of S_{i} in
the expression f
.
Dom::ImageSet(f, x, S)
represents the set
. Dom::ImageSet(f,
[x1, ..., xn], [S1, ..., Sn])
represents the set
.
f
need not contain x
;
on the other hand, it may contain other identifiers (free variables).
If a list of several identifiers is given, the identifiers must be distinct.
S
must be a set; see solve
for an overview of the different
kinds of sets in MuPAD^{®}.
Dom::ImageSet
carries out some automatical
simplifications that may produce a result of a type different from Dom::ImageSet
.
Dom::ImageSet
renames the variables x1,
..., xn
, in order to avoid naming conflicts as well as producing
a nicer output.
We define S to be the set of all integer multiples of π.
S:= Dom::ImageSet(ugly*PI, ugly, Z_)
Our ugly variable name has been replaced by a nicer one which suggests that it represents an integer.
We may now apply the usual settheoretic operations.
S intersect Dom::Interval(3..7)
An element of an image set may be obtained by substituting all parameters by some values:
S:= Dom::ImageSet(a^7 + b^3 + C, [a, b], [Z_, Z_])
On calling the evalParam
method, we have
to take care that the variable names have been replaced.
Dom::ImageSet::evalParam(S, k = 3, l = 5)
The same may be achieved using the index operator:
S[3, 5]
Substituting only for one parameter, we obtain an image set in the other parameter:
Dom::ImageSet::evalParam(S, k = 3)
A parameter may be substituted by itself, meaning that it becomes a free variable:
Dom::ImageSet::evalParam(S, k = k)
The evalParam
method cannot be used to substitute
a free variable:
Dom::ImageSet::evalParam(S, C = 3)
delete S:

Arithmetical expression 

Identifier or indexed identifier 

Set of any type 