### 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 4.12.

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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