GAP (ref) - Chapter 20: Booleans

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20 Booleans
20.1 IsBool (Filter)
20.1-1 IsBool
20.2 Fail (Variable)
20.2-1 fail
20.3 Comparisons of Booleans
20.3-1 Equality and inequality of Booleans
20.3-2 Ordering of Booleans
20.4 Operations for Booleans
20.4-1 Logical disjunction
20.4-2 Logical conjunction
20.4-3 Logical negation

### 20 Booleans

The two main boolean values are `true` and `false`. They stand for the logical values of the same name. They appear as values of the conditions in `if`-statements and `while`-loops. Booleans are also important as return values of filters (see 13.2) such as `IsFinite` (30.4-2) and `IsBool` (20.1-1). Note that it is a convention that the name of a function that returns `true` or `false` according to the outcome, starts with `Is`.

For technical reasons, also the value `fail` (see 20.2) is regarded as a boolean.

#### 20.1 IsBool (Filter)

##### 20.1-1 IsBool

`‣ IsBool`( obj ) | ( category )

tests whether obj is `true`, `false` or `fail`.

    gap> IsBool( true );  IsBool( false );  IsBool( 17 );
    true
    true
    false

#### 20.2 Fail (Variable)

##### 20.2-1 fail

`‣ fail` | ( global variable )

The value `fail` is used to indicate situations when an operation could not be performed for the given arguments, either because of shortcomings of the arguments or because of restrictions in the implementation or computability. So for example `Position` (21.16-1) will return `fail` if the point searched for is not in the list.

`fail` is simply an object that is different from every other object than itself.

For technical reasons, `fail` is a boolean value. But note that `fail` cannot be used to form boolean expressions with `and`, `or`, and `not` (see 20.4 below), and `fail` cannot appear in boolean lists (see Chapter 22).

#### 20.3 Comparisons of Booleans

##### 20.3-1 Equality and inequality of Booleans

`bool1 = bool2`

`bool1 <> bool2`

The equality operator `=` evaluates to `true` if the two boolean values bool1 and bool2 are equal, i.e., both are `true` or both are `false` or both `fail`, and `false` otherwise. The inequality operator `<>` evaluates to `true` if the two boolean values bool1, bool2 are different, and `false` otherwise. This operation is also called the exclusive or, because its value is `true` if exactly one of bool1 or bool2 is `true`.

You can compare boolean values with objects of other types. Of course they are never equal.

    gap> true = false;
    false
    gap> false = (true = fail);
    true
    gap> true <> 17;
    true

##### 20.3-2 Ordering of Booleans

bool1 `<` bool2

The ordering of boolean values is defined by `true` `<` `false` `<` `fail`. For the comparison of booleans with other GAP objects, see Section 4.13.

    gap> true < false;  fail >= false;
    true
    true

#### 20.4 Operations for Booleans

The following boolean operations are only applicable to `true` and `false`.

##### 20.4-1 Logical disjunction

bool1 `or` bool2

The logical operator `or` evaluates to `true` if at least one of the two boolean operands bool1 and bool2 is `true`, and to `false` otherwise.

`or` first evaluates bool1. If the value is neither `true` nor `false` an error is signalled. If the value is `true`, then `or` returns `true` without evaluating bool2. If the value is `false`, then `or` evaluates bool2. Again, if the value is neither `true` nor `false` an error is signalled. Otherwise `or` returns the value of bool2. This short-circuited evaluation is important if the value of bool1 is `true` and evaluation of bool2 would take much time or cause an error.

`or` is associative, i.e., it is allowed to write b1 `or` b2 `or` b3, which is interpreted as (b1 `or` b2) `or` b3. `or` has the lowest precedence of the logical operators. All logical operators have lower precedence than the comparison operators `=`, `<`, `in`, etc.

    gap> true or false;
    true
    gap> false or false;
    false
    gap> i := -1;; l := [1,2,3];;
    gap> if i <= 0 or l[i] = false then     # this does not cause an error,
    >    Print("aha\n"); fi;                # because `l[i]' is not evaluated
    aha

##### 20.4-2 Logical conjunction

bool1 `and` bool2

fil1 `and` fil2

The logical operator `and` evaluates to `true` if both boolean operands bool1, bool2 are `true`, and to `false` otherwise.

`and` first evaluates bool1. If the value is neither `true` nor `false` an error is signalled. If the value is `false`, then `and` returns `false` without evaluating bool2. If the value is `true`, then `and` evaluates bool2. Again, if the value is neither `true` nor `false` an error is signalled. Otherwise `and` returns the value of bool2. This short-circuited evaluation is important if the value of bool1 is `false` and evaluation of bool2 would take much time or cause an error.

`and` is associative, i.e., it is allowed to write b1 `and` b2 `and` b3, which is interpreted as (b1 `and` b2) `and` b3. `and` has higher precedence than the logical `or` operator, but lower than the unary logical `not` operator. All logical operators have lower precedence than the comparison operators `=`, `<`, `in`, etc.

    gap> true and false;
    false
    gap> true and true;
    true
    gap> false and 17;  # does not cause error, because 17 is never looked at
    false

`and` can also be applied to filters. It returns a filter that when applied to some argument x, tests fil1\((x)\) `and` fil2\((x)\).

    gap> andfilt:= IsPosRat and IsInt;;
    gap> andfilt( 17 );  andfilt( 1/2 );
    true
    false

##### 20.4-3 Logical negation

`not` bool

The logical operator `not` returns `true` if the boolean value bool is `false`, and `true` otherwise. An error is signalled if bool does not evaluate to `true` or `false`.

`not` has higher precedence than the other logical operators, `or` and `and`. All logical operators have lower precedence than the comparison operators `=`, `<`, `in`, etc.

    gap> true and false;
    false
    gap> not true;
    false
    gap> not false;
    true

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