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<a name="Element_002dby_002delement-Boolean-Operators"></a>
<div class="header">
<p>
Next: <a href="Short_002dcircuit-Boolean-Operators.html#Short_002dcircuit-Boolean-Operators" accesskey="n" rel="next">Short-circuit Boolean Operators</a>, Up: <a href="Boolean-Expressions.html#Boolean-Expressions" accesskey="u" rel="up">Boolean Expressions</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Element_002dby_002delement-Boolean-Operators-1"></a>
<h4 class="subsection">8.5.1 Element-by-element Boolean Operators</h4>
<a name="index-element_002dby_002delement-evaluation"></a>
<p>An <em>element-by-element boolean expression</em> is a combination of
comparison expressions using the boolean
operators “or” (‘<samp>|</samp>’), “and” (‘<samp>&</samp>’), and “not” (‘<samp>!</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>
<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>
<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>
<p>Here are descriptions of the three element-by-element boolean operators.
</p>
<dl compact="compact">
<dt><code><var>boolean1</var> & <var>boolean2</var></code></dt>
<dd><a name="index-_0026"></a>
<p>Elements of the result are true if both corresponding elements of
<var>boolean1</var> and <var>boolean2</var> are true.
</p>
</dd>
<dt><code><var>boolean1</var> | <var>boolean2</var></code></dt>
<dd><a name="index-_007c"></a>
<p>Elements of the result are true if either of the corresponding elements
of <var>boolean1</var> or <var>boolean2</var> is true.
</p>
</dd>
<dt><code>! <var>boolean</var></code></dt>
<dt><code>~ <var>boolean</var></code></dt>
<dd><a name="index-_007e"></a>
<a name="index-_0021"></a>
<p>Each element of the result is true if the corresponding element of
<var>boolean</var> is false.
</p></dd>
</dl>
<p>These operators work on an element-by-element basis. For example, the
expression
</p>
<div class="example">
<pre class="example">[1, 0; 0, 1] & [1, 0; 2, 3]
</pre></div>
<p>returns a two by two identity matrix.
</p>
<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>
<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
</p>
<div class="example">
<pre class="example">a & b++
</pre></div>
<p>the value of the variable <var>b</var> is incremented even if the variable
<var>a</var> is zero.
</p>
<p>This behavior is necessary for the boolean operators to work as
described for matrix-valued operands.
</p>
<a name="index-_0026-1"></a>
<a name="XREFand"></a><dl>
<dt><a name="index-and"></a>Built-in Function: <em></em> <strong>and</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-and-1"></a>Built-in Function: <em></em> <strong>and</strong> <em>(<var>x1</var>, <var>x2</var>, …)</em></dt>
<dd><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:
</p>
<div class="example">
<pre class="example">(…((x1 & x2) & x3) & …)
</pre></div>
<p>At least one argument is required.
</p>
<p><strong>See also:</strong> <a href="#XREFor">or</a>, <a href="#XREFnot">not</a>, <a href="Finding-Elements-and-Checking-Conditions.html#XREFxor">xor</a>.
</p></dd></dl>
<a name="index-_007e-1"></a>
<a name="index-_0021-1"></a>
<a name="XREFnot"></a><dl>
<dt><a name="index-not"></a>Built-in Function: <em></em> <strong>not</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Return the logical NOT of <var>x</var>. This function is equivalent to
<code>! x</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFand">and</a>, <a href="#XREFor">or</a>, <a href="Finding-Elements-and-Checking-Conditions.html#XREFxor">xor</a>.
</p></dd></dl>
<a name="index-_007c-1"></a>
<a name="XREFor"></a><dl>
<dt><a name="index-or"></a>Built-in Function: <em></em> <strong>or</strong> <em>(<var>x</var>, <var>y</var>)</em></dt>
<dt><a name="index-or-1"></a>Built-in Function: <em></em> <strong>or</strong> <em>(<var>x1</var>, <var>x2</var>, …)</em></dt>
<dd><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:
</p>
<div class="example">
<pre class="example">(…((x1 | x2) | x3) | …)
</pre></div>
<p>At least one argument is required.
</p>
<p><strong>See also:</strong> <a href="#XREFand">and</a>, <a href="#XREFnot">not</a>, <a href="Finding-Elements-and-Checking-Conditions.html#XREFxor">xor</a>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="Short_002dcircuit-Boolean-Operators.html#Short_002dcircuit-Boolean-Operators" accesskey="n" rel="next">Short-circuit Boolean Operators</a>, Up: <a href="Boolean-Expressions.html#Boolean-Expressions" accesskey="u" rel="up">Boolean Expressions</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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