bool
Boolean evaluation
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For booleans in MATLAB^{®}, see logical
.
bool(b
)
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 nonzero real or complex floatingpoint 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
.
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 floatingpoint approximations. Alternatively,
you can use is
:
bool(float(sqrt(14)) <= float(sqrt(2)*sqrt(7))), is(sqrt(14) <= sqrt(2)*sqrt(7))
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)
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)
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:
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:
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
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 
TRUE
, FALSE
, or UNKNOWN
.
b