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<h4 class="subsection">8.5.1 Element-by-element Boolean Operators</h4>
<p><a name="index-element_002dby_002delement-evaluation-518"></a>
An <dfn>element-by-element boolean expression</dfn> is a combination of
comparison expressions using the boolean
operators “or” (‘<samp><span class="samp">|</span></samp>’), “and” (‘<samp><span class="samp">&</span></samp>’), and “not” (‘<samp><span class="samp">!</span></samp>’),
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.
<p>Element-by-element boolean expressions can be used wherever comparison
expressions can be used. They can be used in <code>if</code> and <code>while</code>
statements. However, a matrix value used as the condition in an
<code>if</code> or <code>while</code> statement is only true if <em>all</em> of its
elements are nonzero.
<p>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.
<p>Here are descriptions of the three element-by-element boolean operators.
<dl>
<dt><var>boolean1</var><code> & </code><var>boolean2</var><dd><a name="index-g_t_0026-519"></a>Elements of the result are true if both corresponding elements of
<var>boolean1</var> and <var>boolean2</var> are true.
<br><dt><var>boolean1</var><code> | </code><var>boolean2</var><dd><a name="index-g_t_007c-520"></a>Elements of the result are true if either of the corresponding elements
of <var>boolean1</var> or <var>boolean2</var> is true.
<br><dt><code>! </code><var>boolean</var><dt><code>~ </code><var>boolean</var><dd><a name="index-g_t_007e-521"></a><a name="index-g_t_0021-522"></a>Each element of the result is true if the corresponding element of
<var>boolean</var> is false.
</dl>
<p>For matrix operands, these operators work on an element-by-element
basis. For example, the expression
<pre class="example"> [1, 0; 0, 1] & [1, 0; 2, 3]
</pre>
<p class="noindent">returns a two by two identity matrix.
<p>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.
<p>For the binary element-by-element boolean operators, both subexpressions
<var>boolean1</var> and <var>boolean2</var> are evaluated before computing the
result. This can make a difference when the expressions have side
effects. For example, in the expression
<pre class="example"> a & b++
</pre>
<p class="noindent">the value of the variable <var>b</var> is incremented even if the variable
<var>a</var> is zero.
<p>This behavior is necessary for the boolean operators to work as
described for matrix-valued operands.
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