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<h4 class="subsection">8.5.1 Element-by-element Boolean Operators</h4>
<p><a name="index-element_002dby_002delement-evaluation-638"></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_0026-639"></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-640"></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-641"></a><a name="index-g_t_0021-642"></a>Each element of the result is true if the corresponding element of
<var>boolean</var> is false.
</dl>
<p>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, broadcasting rules apply. See <a href="Broadcasting.html#Broadcasting">Broadcasting</a>.
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.
<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.
<p><a name="index-g_t_0026-643"></a><!-- and src/data.cc -->
<a name="doc_002dand"></a>
<div class="defun">
— Built-in Function: <b>and</b> (<var>x, y</var>)<var><a name="index-and-644"></a></var><br>
— Built-in Function: <b>and</b> (<var>x1, x2, <small class="dots">...</small></var>)<var><a name="index-and-645"></a></var><br>
<blockquote><p>Return the logical AND of <var>x</var> and <var>y</var>.
This function is equivalent to <code>x & y</code><!-- /@w -->.
If more arguments are given, the logical and is applied
cumulatively from left to right:
<pre class="example"> (...((x1 & x2) & x3) & ...)
</pre>
<p>At least one argument is required.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dor.html#doc_002dor">or</a>, <a href="doc_002dnot.html#doc_002dnot">not</a>, <a href="doc_002dxor.html#doc_002dxor">xor</a>.
</p></blockquote></div>
<p><a name="index-g_t_007e-646"></a><a name="index-g_t_0021-647"></a><!-- not src/data.cc -->
<a name="doc_002dnot"></a>
<div class="defun">
— Built-in Function: <b>not</b> (<var>x</var>)<var><a name="index-not-648"></a></var><br>
<blockquote><p>Return the logical NOT of <var>x</var>. This function is equivalent to
<code>! x</code>.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dand.html#doc_002dand">and</a>, <a href="doc_002dor.html#doc_002dor">or</a>, <a href="doc_002dxor.html#doc_002dxor">xor</a>.
</p></blockquote></div>
<p><a name="index-g_t_007c-649"></a><!-- or src/data.cc -->
<a name="doc_002dor"></a>
<div class="defun">
— Built-in Function: <b>or</b> (<var>x, y</var>)<var><a name="index-or-650"></a></var><br>
— Built-in Function: <b>or</b> (<var>x1, x2, <small class="dots">...</small></var>)<var><a name="index-or-651"></a></var><br>
<blockquote><p>Return the logical OR of <var>x</var> and <var>y</var>.
This function is equivalent to <code>x | y</code><!-- /@w -->.
If more arguments are given, the logical or is applied
cumulatively from left to right:
<pre class="example"> (...((x1 | x2) | x3) | ...)
</pre>
<p>At least one argument is required.
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dand.html#doc_002dand">and</a>, <a href="doc_002dnot.html#doc_002dnot">not</a>, <a href="doc_002dxor.html#doc_002dxor">xor</a>.
</p></blockquote></div>
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