in, _in

Membership

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Syntax

x in set
_in(x, set)
for y in object do body end_for
f(y) $ y in object

Description

x in set is the MuPAD® notation for the statement “x is a member of set.”

In conjunction with one of the keywords for or $, the meaning changes to “iterate over all operands of the object”. See for and $ for details. Cf. Example 6.

Apart from the usage with for and $, the statement x in object is equivalent to the function call _in(x, object).

x in set is just evaluated to itself. expand(x in set) tries to return an equivalent expression without using the operator in, as described in the following paragraphs.

For sets of type DOM_SET, set unions, differences and intersections, x in set is expanded to an equivalent Boolean expression of equations and expressions involving in. Cf. Example 1.

If set is a solution set of a single equation in one unknown, given by a symbolic call to solve, expanding in returns a Boolean condition that is equivalent to x being a solution. Cf. Example 2.

If set is a RootOf expression, expanding in returns a Boolean condition that is equivalent to x being a root of the corresponding equation. Cf. Example 3.

The function bool and every function that uses boolean evaluation can also handle many logical expressions involving in. Cf. Example 4.

The function is handles various logical statements involving in, including a variety of types for the parameter set which are not handled by in itself. Cf. Example 5 for a few typical cases.

Apart from the usual overloading mechanism by the first argument of an in call, in can be overloaded by its second argument, too. This argument must define the slot"set2expr" for this purpose. The slot will be called with the arguments set, x.

Examples

Example 1

x in {1, 2, 3} is expanded into an equivalent statement involving = and or:

expand(x in {1, 2, 3})

The same happens if you replace x by a number, because Boolean expressions are only evaluated inside certain functions such as bool or is:

expand(1 in {1, 2, 3}), bool(1 in {1, 2, 3}), is(1 in {1, 2, 3})

If only some part of the expression can be simplified this way, the returned expression can contain unevaluated calls to in:

expand(x in {1, 2, 3} union A)

Example 2

For symbolic calls to solve representing the solution set of a single equation in one unknown, in can be used to check whether a particular value lies in the solution set:

solve(cos(x) + x = cos(2) + 2, x);
expand(2 in %), bool(2 in %)

Example 3

in can be used to check whether a value is a member of the solution set represented by a RootOf expression:

r := RootOf(x^2 - 1, x);
expand(1 in r), bool(1 in r), expand(2 in r), bool(2 in r)

expand((y - 1) in RootOf(x^2 - 1 - y^2 + 2*y, x))

expand(%)

delete r:

Example 4

Expressions with operator in are boolean expressions: they can be used like equations or inequalities.

if 2 in {2, 3, 5} then "ok" end

Example 5

The MuPAD function is can investigate membership of objects in infinite sets. It respects properties of identifiers:

is(123 in Q_), is(2/3 in Q_)

Example 6

In conjunction with for and $, y in object iterates y over all operands of the object:

for y in [1, 2] do
   print(y)
end_for:

y^2 + 1 $ y in a + b*c + d^2

delete y:

Parameters

x

An arbitrary MuPAD object

set

A set or an object of set-like type

y

An identifier or a local variable (DOM_VAR) of a procedure

object, f(y)

Arbitrary MuPAD objects

Return Values

x in set just returns the input.

Overloaded By

set, x

See Also

MuPAD Functions