#### 10.5.1 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, if 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.