Documentation

# `solvelib`::`isEmpty`

Predicate expressing the emptyness of a set

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## Syntax

```solvelib::isEmpty(`S`)
```

## Description

`solvelib::isEmpty(S)` returns a boolean expression that is equivalent to the statement that `S` is the empty set.

Since functions operating on boolean expressions like `assume`, `is`, or `solve` cannot handle equations involving sets, it is not possible to pass the expression `S={}` to them. `solvelib::isEmpty(S)` helps to get around this problem, as it tries to express the emptyness of `S` in an equivalent way that can be handled by the aforementioned functions. If no suitable equivalen expression is found, the unevaluated call to `solvelib::isEmpty` is returned.

`S` may be a set of any type; see `solve` for an enumeration of the various types of sets.

The `solvelib::isEmpty` function always returns Boolean expressions, even if the function cannot resolve an expression. See Example 4.

## Examples

### Example 1

The emptyness of a `DOM_SET` can be decided immediately:

`solvelib::isEmpty({a, b}), solvelib::isEmpty({})`
` `

### Example 2

The intersection of a finite set with any other set is empty if and only if none of the elements of the finite set is in the other set:

`solvelib::isEmpty({a, b} intersect Z_)`
` `

### Example 3

The output of `solve` can be entered directly into `solvelib::isEmpty`:

`solvelib::isEmpty(solve(a*x=b, x))`
` `

### Example 4

Sometimes, no simpler equivalent expression can be found:

`result := solvelib::isEmpty(solve(x^2 = sin(x), x))`
` `

The returned expression is a Boolean expression:

`testtype(result, Type::Boolean)`
` `

## Parameters

 `S` Any set

## Return Values

Boolean expression

`S`