An *element-by-element boolean expression* is a combination of
comparison expressions using the boolean
operators “or” (‘`|`’), “and” (‘`&`’), and “not” (‘`!`’),
along with parentheses to control nesting. The truth of the boolean
expression is computed by combining the truth values of the
corresponding elements of the component expressions. A value is
considered to be false if it is zero, and true otherwise.

Element-by-element boolean expressions can be used wherever comparison
expressions can be used. They can be used in `if`

and `while`

statements. However, a matrix value used as the condition in an
`if`

or `while`

statement is only true if *all* of its
elements are nonzero.

Like comparison operations, each element of an element-by-element boolean expression also has a numeric value (1 if true, 0 if false) that comes into play if the result of the boolean expression is stored in a variable, or used in arithmetic.

Here are descriptions of the three element-by-element boolean operators.

¶`boolean1`&`boolean2`Elements of the result are true if both corresponding elements of

`boolean1`and`boolean2`are true.

¶`boolean1`|`boolean2`Elements of the result are true if either of the corresponding elements of

`boolean1`or`boolean2`is true.`!`

¶`boolean``~`

`boolean`-
Each element of the result is true if the corresponding element of

`boolean`is false.

These operators work on an element-by-element basis. For example, the expression

[1, 0; 0, 1] & [1, 0; 2, 3]

returns a two by two identity matrix.

For the binary operators, broadcasting rules apply. See Broadcasting. In particular, if one of the operands is a scalar and the other a matrix, the operator is applied to the scalar and each element of the matrix.

For the binary element-by-element boolean operators, both subexpressions
`boolean1` and `boolean2` are evaluated before computing the
result. This can make a difference when the expressions have side
effects. For example, in the expression

a & b++

the value of the variable `b` is incremented even if the variable
`a` is zero.

This behavior is necessary for the boolean operators to work as described for matrix-valued operands.

- :
`TF`=**and**`(`

¶`x`,`y`) - :
`TF`=**and**`(`

¶`x1`,`x2`, …) Return the logical AND of

`x`and`y`.This function is equivalent to the operator syntax

. If more than two arguments are given, the logical AND is applied cumulatively from left to right:`x`&`y`(…((

`x1`&`x2`) &`x3`) & …)

- :
`z`=**not**`(`

¶`x`) Return the logical NOT of

`x`.This function is equivalent to the operator syntax

`!`

.`x`