# `<=>`, `_equiv`

Logical equivalence

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## Syntax

```b1 <=> b2
_equiv(`b1`, `b2`)
```

## Description

`b1 <=> b2` represents the logical equivalence of the Boolean expressions `b1`, `b2`.

`a <=> b` is equivalent to ```(a ==> b) and (b ==> a)```, where `a ==> b` is equivalent to `not a or b`.

`_equiv(a, b)` is equivalent to ```a <=> b```.

MuPAD® uses a three state logic with the Boolean constants `TRUE`, `FALSE`, and `UNKNOWN`. These are processed as follows:

 `and` `TRUE` `FALSE` `UNKNOWN` `TRUE` `TRUE` `FALSE` `UNKNOWN` `FALSE` `FALSE` `FALSE` `FALSE` `UNKNOWN` `UNKNOWN` `FALSE` `UNKNOWN`

Boolean expressions can be composed of these constants as well as of arbitrary arithmetical expressions. Typically, equations, such as `x = y`, and inequalities, such as ```x <> y```, `x < y`, ```x <= y```, are used to construct Boolean expressions.

Combinations of the constants `TRUE`, `FALSE`, `UNKNOWN` inside a Boolean expression are simplified automatically. However, symbolic Boolean subexpressions, equalities, and inequalities are not evaluated and simplified by logical operators. Use `bool` to evaluate such expressions to one of the Boolean constants. Note, however, that `bool` can evaluate inequalities ```x < y```, `x <= y` and so on, only if they are composed of numbers of type `Type::Real`.

Use `simplify` with the option `logic` to simplify expressions involving symbolic Boolean subexpressions. See Example 1.

The precedences of the logical operators are as follows. If in doubt, use parentheses to ensure that the expression is parsed as desired.

• The operator `not` is stronger binding than `and`, that is, `not b1 and b2` = ```(not b1) and b2```.

• The operator `and` is stronger binding than `xor`, that is, `b1 and b2 or b3` = ```(b1 and b2) xor b3```.

• The operator `xor` is stronger binding than `or`, that is, `b1 xor b2 or b3` = ```(b1 xor b2) or b3```.

• The operator `or` is stronger binding than `==>`, that is, ```b1 or b2 ==> b3``` = `(b1 or b2) ==> b3`.

• The operator `==>` is stronger binding than `<=>`, that is, ```b1 ==> b2 <=> b3``` = `(b1 ==> b2) <=> b3`.

In the conditional context of `if`, `repeat`, and `while` statements, Boolean expressions are evaluated via “lazy evaluation” (see `_lazy_and`, `_lazy_or`). In any other context, all operands are evaluated.

## Examples

### Example 1

`a <=> b` is equivalent to ```(a ==> b) and (b ==> a)```, where `a ==> b` is equivalent to `not a or b`.

`simplify((a <=> b), logic)`

`simplify((a ==> b) and (b ==> a), logic)`

## Parameters

 `b1, b2` Boolean expressions

## Return Values

Boolean expression.

`b`, `b_1`, `b_2`