Logical equivalence
This functionality does not run in MATLAB.
b_{1} <=> b_{2}
_equiv(b_{1}
,b_{2}
)
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.
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)

Boolean expressions 
Boolean expression.
b
, b_1
, b_2