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and, _and

Logical “and”

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b1 and b2
_and(b1, b2, …)


b1 and b2 represents the logical and of the Boolean expressions b1, b2.

MuPAD® uses a three state logic with the Boolean constants TRUE, FALSE, and UNKNOWN. These are processed as follows:


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, and x <= y, are used to construct Boolean expressions.

_and(b1, b2, ...) is equivalent to b1 and b2 and .... This expression represents TRUE if every single expression evaluates to TRUE. It represents FALSE if at least one expression evaluates to FALSE. It represents UNKNOWN if at least one expression evaluates to UNKNOWN and all others evaluate to TRUE.

_and() returns TRUE.

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. See Example 2.

Use simplify with the option logic to simplify expressions involving symbolic Boolean subexpressions. See Example 3.

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.


Example 1

Combinations of the Boolean constants TRUE, FALSE, and UNKNOWN are simplified automatically to one of the constants:

TRUE and not (FALSE or TRUE)


Example 2

Logical operators simplify subexpressions that evaluate to the constants TRUE, FALSE, UNKNOWN.

b1 or b2 and TRUE

FALSE or ((not b1) and TRUE)

b1 and (b2 or FALSE) and UNKNOWN

FALSE or (b1 and UNKNOWN) or x < 1

TRUE and ((b1 and FALSE) or (b1 and TRUE))

However, equalities and inequalities are not evaluated:

(x = x) and (1 < 2) and (2 < 3) and (3 < 4)

Boolean evaluation is enforced via bool:


Example 3

Expressions involving symbolic Boolean subexpressions are not simplified by and, or, not. Simplification has to be requested explicitly via the function simplify:

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

simplify(%, logic)

Example 4

The Boolean functions _and and _or accept arbitrary sequences of Boolean expressions. The following call uses isprime to check whether all elements of the given set are prime:

set := {1987, 1993, 1997, 1999, 2001}: 
_and(isprime(i) $ i in set)

The following call checks whether at least one of the numbers is prime:

_or(isprime(i) $ i in set)

delete set:


b1, b2, …

Boolean expressions

Return Values

Boolean expression.

Overloaded By

b, b_1, b_2

See Also

MuPAD Functions

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