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#### 8.5.1 Element-by-element Boolean Operators

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.

For matrix operands, 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, the dimensions of the operands must conform if both are matrices. 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.

— Built-in Function: and (x, y)
— Built-in Function: and (x1, x2, ...)

> Return the logical AND of x and y. This function is equivalent to `x & y`. If more arguments are given, the logical and is applied cumulatively from left to right:

>                 (...((x1 & x2) & x3) & ...)

> At least one argument is required.

> See also: or, not.

— Built-in Function: not (x)

> Return the logical NOT of x. This function is equivalent to `! x`.

> See also: and, or.

— Built-in Function: or (x, y)
— Built-in Function: or (x1, x2, ...)

> Return the logical OR of x and y. This function is equivalent to `x | y`. If more arguments are given, the logical or is applied cumulatively from left to right:

>                 (...((x1 | x2) | x3) | ...)

> At least one argument is required.

> See also: and, not.