Boolean evaluation

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bool(b) evaluates the Boolean expression b.

The function bool serves for reducing Boolean expressions to one of the Boolean constants TRUE, FALSE, or UNKNOWN.

Boolean expressions are expressions that are composed of equalities, inequalities, elementhood relations, and these constants, combined via the logical operators and, or, not.

The function bool evaluates all equalities and inequalities inside a Boolean expression to either TRUE or FALSE. The resulting logical combination of the Boolean constants is reduced according to the rules of the MuPAD® three state logic (see and, or, not).


Equations x = y and inequalities x <> y are evaluated syntactically by bool. It does not test equality in any mathematical sense.


Inequalities x < y, x <= y etc. can be evaluated by bool if and only if x and y are real numbers of type Type::Real. Otherwise, an error occurs.

bool evaluates all subexpressions of a Boolean expression before simplifying the result. The functions _lazy_and, _lazy_or provide an alternative: “lazy Boolean evaluation”.

There is no need to use bool in the conditional part of if, repeat, and while statements. Internally, these statements enforce Boolean evaluation by _lazy_and and _lazy_or. Cf. Example 5.

Use simplify with the option logic to simplify expressions involving symbolic Boolean subexpressions. Cf. Example 7.

bool is overloadable not only for domains, but also for function environments. This means that, if f evaluates to a function environment, then bool(f(x1, …, xn)) returns f::bool( x1, …, xn ), or an error if no slot f::bool exists.

The call bool(x ~= y) serves for comparing numerical values x and y. If both x and y can be converted to non-zero real or complex floating-point numbers, it is checked whether float((x - y)/x)| < 10^(-DIGITS) is satisfied. Thus, TRUE is returned if x and y coincide within the relative numerical precision set by DIGITS. For x = 0, the criterion is |float(y)| < 10^(-DIGITS). For y = 0, the criterion is |float(x)| < 10^(-DIGITS). If either x or y contains a symbolic object that cannot be converted to a real or complex floating point number, the function bool returns the value UNKNOWN.


Example 1

MuPAD realizes that 1 is less than 2:

1 < 2 = bool(1 < 2)

Note that bool can fail to compare real numbers expressed symbolically:

bool(sqrt(14) <= sqrt(2)*sqrt(7))
Error: Unable to evaluate to Boolean. [_leequal]

You can compare floating-point approximations. Alternatively, you can use is:

bool(float(sqrt(14)) <= float(sqrt(2)*sqrt(7))),
is(sqrt(14) <= sqrt(2)*sqrt(7))

Example 2

The Boolean operators and, or, not do not evaluate equations and inequalities logically, and return a symbolic Boolean expression. Boolean evaluation and simplification is enforced by bool:

a = a and 3 < 4

bool(a = a and 3 < 4)

Example 3

bool handles the special Boolean constant UNKNOWN:

bool(UNKNOWN and 1 < 2), bool(UNKNOWN or 1 < 2),
bool(UNKNOWN and 1 > 2), bool(UNKNOWN or 1 > 2)

Example 4

bool must be able to reduce all parts of a composite Boolean expression to one of the Boolean constants. No symbolic Boolean subexpressions may be involved:

b := b1 and b2 or b3: bool(b)
Error: Unable to evaluate to Boolean. [bool]
b1 := 1 < 2: b2 := x = x: b3 := FALSE: bool(b)

delete b, b1, b2, b3:

Example 5

There is no need to use bool explicitly in the conditional parts of if, repeat, and while statements. Note, however, that these structures internally use “lazy evaluation” via _lazy_and and _lazy_or rather than “complete Boolean evaluation” via bool:

x := 0: if x <> 0 and sin(1/x) = 0 then 1 else 2 end

In contrast to “lazy evaluation”, bool evaluates all conditions. Consequently, a division by zero occurs in the evaluation of sin(1/x) = 0:

bool(x <> 0 and sin(1/x) = 0)
Error: Division by zero. [_invert]
delete x:

Example 6

Note that bool does not operate recursively. The following calls are completely different, the first one comparing the expression TRUE = TRUE and the constant TRUE (syntactically), the second one comparing the result of another bool-call with TRUE:

bool((TRUE = TRUE) = TRUE);
bool(bool(TRUE = TRUE) = TRUE)

Since if, while and similar constructs use the same Boolean evaluation internally, this also effects conditions in such clauses:

if (is(a < b) = TRUE) or (3 = 3) then YES else NO end;
if (is(a < b) or (3 = 3)) = TRUE then YES else NO end

Example 7

Expressions involving symbolic Boolean subexpressions cannot be processed by bool. However, simplify with the option logic can be used for simplification:

(b1 and b2) or (b1 and (not b2)) and (1 < 2)

simplify(%, logic)



A Boolean expression

Return Values


Overloaded By


See Also

MuPAD Functions