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<div class="section" id="Set-Operations">
<div class="header">
<p>
Up: <a href="Sets.html" accesskey="u" rel="up">Sets</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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<hr>
<span id="Set-Operations-1"></span><h3 class="section">27.1 Set Operations</h3>
<p>Octave supports several basic set operations. Octave can compute the union,
intersection, and difference of two sets. Octave also supports the
<em>Exclusive Or</em> set operation.
</p>
<p>The functions for set operations all work in the same way by accepting two
input sets and returning a third set. As an example, assume that <code>a</code> and
<code>b</code> contains two sets, then
</p>
<div class="example">
<pre class="example">union (a, b)
</pre></div>
<p>computes the union of the two sets.
</p>
<p>Finally, determining whether elements belong to a set can be done with the
<code>ismember</code> function. Because sets are ordered this operation is very
efficient and is of order O(log2(n)) which is preferable to the <code>find</code>
function which is of order O(n).
</p>
<span id="XREFintersect"></span><dl class="def">
<dt id="index-intersect"><span class="category">: </span><span><em><var>c</var> =</em> <strong>intersect</strong> <em>(<var>a</var>, <var>b</var>)</em><a href='#index-intersect' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-intersect-1"><span class="category">: </span><span><em><var>c</var> =</em> <strong>intersect</strong> <em>(<var>a</var>, <var>b</var>, "rows")</em><a href='#index-intersect-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-intersect-2"><span class="category">: </span><span><em><var>c</var> =</em> <strong>intersect</strong> <em>(…, "sorted")</em><a href='#index-intersect-2' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-intersect-3"><span class="category">: </span><span><em><var>c</var> =</em> <strong>intersect</strong> <em>(…, "stable")</em><a href='#index-intersect-3' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-intersect-4"><span class="category">: </span><span><em><var>c</var> =</em> <strong>intersect</strong> <em>(…, "legacy")</em><a href='#index-intersect-4' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-intersect-5"><span class="category">: </span><span><em>[<var>c</var>, <var>ia</var>, <var>ib</var>] =</em> <strong>intersect</strong> <em>(…)</em><a href='#index-intersect-5' class='copiable-anchor'> ¶</a></span></dt>
<dd>
<p>Return the unique elements common to both <var>a</var> and <var>b</var>.
</p>
<p>If <var>a</var> and <var>b</var> are both row vectors then return a row vector;
Otherwise, return a column vector. The inputs may also be cell arrays of
strings.
</p>
<p>If the optional input <code>"rows"</code> is given then return the common rows of
<var>a</var> and <var>b</var>. The inputs must be 2-D numeric matrices to use this
option.
</p>
<p>The optional argument <code>"sorted"</code>/<code>"stable"</code> controls the order
in which unique values appear in the output. The default is
<code>"sorted"</code> and values in the output are placed in ascending order.
The alternative <code>"stable"</code> preserves the order found in the input.
</p>
<p>If requested, return column index vectors <var>ia</var> and <var>ib</var> such that
<code><var>c</var> = <var>a</var>(<var>ia</var>)</code> and <code><var>c</var> = <var>b</var>(<var>ib</var>)</code>.
</p>
<p>Programming Note: The input flag <code>"legacy"</code> changes the algorithm
to be compatible with <small>MATLAB</small> releases prior to R2012b.
</p>
<p><strong>See also:</strong> <a href="Sets.html#XREFunique">unique</a>, <a href="#XREFunion">union</a>, <a href="#XREFsetdiff">setdiff</a>, <a href="#XREFsetxor">setxor</a>, <a href="#XREFismember">ismember</a>.
</p></dd></dl>
<span id="XREFunion"></span><dl class="def">
<dt id="index-union"><span class="category">: </span><span><em><var>c</var> =</em> <strong>union</strong> <em>(<var>a</var>, <var>b</var>)</em><a href='#index-union' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-union-1"><span class="category">: </span><span><em><var>c</var> =</em> <strong>union</strong> <em>(<var>a</var>, <var>b</var>, "rows")</em><a href='#index-union-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-union-2"><span class="category">: </span><span><em><var>c</var> =</em> <strong>union</strong> <em>(…, "sorted")</em><a href='#index-union-2' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-union-3"><span class="category">: </span><span><em><var>c</var> =</em> <strong>union</strong> <em>(…, "stable")</em><a href='#index-union-3' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-union-4"><span class="category">: </span><span><em><var>c</var> =</em> <strong>union</strong> <em>(…, "legacy")</em><a href='#index-union-4' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-union-5"><span class="category">: </span><span><em>[<var>c</var>, <var>ia</var>, <var>ib</var>] =</em> <strong>union</strong> <em>(…)</em><a href='#index-union-5' class='copiable-anchor'> ¶</a></span></dt>
<dd>
<p>Return the unique elements that are in either <var>a</var> or <var>b</var>.
</p>
<p>If <var>a</var> and <var>b</var> are both row vectors then return a row vector;
Otherwise, return a column vector. The inputs may also be cell arrays of
strings.
</p>
<p>If the optional input <code>"rows"</code> is given then return rows that are in
either <var>a</var> or <var>b</var>. The inputs must be 2-D numeric matrices to use
this option.
</p>
<p>The optional argument <code>"sorted"</code>/<code>"stable"</code> controls the order
in which unique values appear in the output. The default is
<code>"sorted"</code> and values in the output are placed in ascending order.
The alternative <code>"stable"</code> preserves the order found in the input.
</p>
<p>The optional outputs <var>ia</var> and <var>ib</var> are column index vectors such
that <code><var>a</var>(<var>ia</var>)</code> and <code><var>b</var>(<var>ib</var>)</code> are disjoint sets
whose union is <var>c</var>.
</p>
<p>Programming Note: The input flag <code>"legacy"</code> changes the algorithm
to be compatible with <small>MATLAB</small> releases prior to R2012b.
</p>
<p><strong>See also:</strong> <a href="Sets.html#XREFunique">unique</a>, <a href="#XREFintersect">intersect</a>, <a href="#XREFsetdiff">setdiff</a>, <a href="#XREFsetxor">setxor</a>, <a href="#XREFismember">ismember</a>.
</p></dd></dl>
<span id="XREFsetdiff"></span><dl class="def">
<dt id="index-setdiff"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setdiff</strong> <em>(<var>a</var>, <var>b</var>)</em><a href='#index-setdiff' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setdiff-1"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setdiff</strong> <em>(<var>a</var>, <var>b</var>, "rows")</em><a href='#index-setdiff-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setdiff-2"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setdiff</strong> <em>(…, "sorted")</em><a href='#index-setdiff-2' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setdiff-3"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setdiff</strong> <em>(…, "stable")</em><a href='#index-setdiff-3' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setdiff-4"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setdiff</strong> <em>(…, "legacy")</em><a href='#index-setdiff-4' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setdiff-5"><span class="category">: </span><span><em>[<var>c</var>, <var>ia</var>] =</em> <strong>setdiff</strong> <em>(…)</em><a href='#index-setdiff-5' class='copiable-anchor'> ¶</a></span></dt>
<dd><p>Return the unique elements in <var>a</var> that are not in <var>b</var>.
</p>
<p>If <var>a</var> is a row vector return a row vector; Otherwise, return a
column vector. The inputs may also be cell arrays of strings.
</p>
<p>If the optional input <code>"rows"</code> is given then return the rows in
<var>a</var> that are not in <var>b</var>. The inputs must be 2-D numeric matrices to
use this option.
</p>
<p>The optional argument <code>"sorted"</code>/<code>"stable"</code> controls the order
in which unique values appear in the output. The default is
<code>"sorted"</code> and values in the output are placed in ascending order.
The alternative <code>"stable"</code> preserves the order found in the input.
</p>
<p>If requested, return the index vector <var>ia</var> such that
<code><var>c</var> = <var>a</var>(<var>ia</var>)</code>.
</p>
<p>Programming Note: The input flag <code>"legacy"</code> changes the algorithm
to be compatible with <small>MATLAB</small> releases prior to R2012b.
</p>
<p><strong>See also:</strong> <a href="Sets.html#XREFunique">unique</a>, <a href="#XREFunion">union</a>, <a href="#XREFintersect">intersect</a>, <a href="#XREFsetxor">setxor</a>, <a href="#XREFismember">ismember</a>.
</p></dd></dl>
<span id="XREFsetxor"></span><dl class="def">
<dt id="index-setxor"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setxor</strong> <em>(<var>a</var>, <var>b</var>)</em><a href='#index-setxor' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setxor-1"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setxor</strong> <em>(<var>a</var>, <var>b</var>, "rows")</em><a href='#index-setxor-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setxor-2"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setxor</strong> <em>(…, "sorted")</em><a href='#index-setxor-2' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setxor-3"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setxor</strong> <em>(…, "stable")</em><a href='#index-setxor-3' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setxor-4"><span class="category">: </span><span><em><var>c</var> =</em> <strong>setxor</strong> <em>(…, "legacy")</em><a href='#index-setxor-4' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-setxor-5"><span class="category">: </span><span><em>[<var>c</var>, <var>ia</var>, <var>ib</var>] =</em> <strong>setxor</strong> <em>(…)</em><a href='#index-setxor-5' class='copiable-anchor'> ¶</a></span></dt>
<dd>
<p>Return the unique elements exclusive to sets <var>a</var> or <var>b</var>.
</p>
<p>If <var>a</var> and <var>b</var> are both row vectors then return a row vector;
Otherwise, return a column vector. The inputs may also be cell arrays of
strings.
</p>
<p>If the optional input <code>"rows"</code> is given then return the rows exclusive
to sets <var>a</var> and <var>b</var>. The inputs must be 2-D numeric matrices to use
this option.
</p>
<p>The optional argument <code>"sorted"</code>/<code>"stable"</code> controls the order
in which unique values appear in the output. The default is
<code>"sorted"</code> and values in the output are placed in ascending order.
The alternative <code>"stable"</code> preserves the order found in the input.
</p>
<p>The optional outputs <var>ia</var> and <var>ib</var> are column index vectors such
that <code><var>a</var>(<var>ia</var>)</code> and <code><var>b</var>(<var>ib</var>)</code> are disjoint sets
whose union is <var>c</var>.
</p>
<p>Programming Note: The input flag <code>"legacy"</code> changes the algorithm
to be compatible with <small>MATLAB</small> releases prior to R2012b.
</p>
<p><strong>See also:</strong> <a href="Sets.html#XREFunique">unique</a>, <a href="#XREFunion">union</a>, <a href="#XREFintersect">intersect</a>, <a href="#XREFsetdiff">setdiff</a>, <a href="#XREFismember">ismember</a>.
</p></dd></dl>
<span id="XREFismember"></span><dl class="def">
<dt id="index-ismember"><span class="category">: </span><span><em><var>tf</var> =</em> <strong>ismember</strong> <em>(<var>a</var>, <var>s</var>)</em><a href='#index-ismember' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-ismember-1"><span class="category">: </span><span><em><var>tf</var> =</em> <strong>ismember</strong> <em>(<var>a</var>, <var>s</var>, "rows")</em><a href='#index-ismember-1' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-ismember-2"><span class="category">: </span><span><em>[<var>tf</var>, <var>s_idx</var>] =</em> <strong>ismember</strong> <em>(…)</em><a href='#index-ismember-2' class='copiable-anchor'> ¶</a></span></dt>
<dd>
<p>Return a logical matrix <var>tf</var> with the same shape as <var>a</var> which is
true (1) if the element in <var>a</var> is found in <var>s</var> and false (0) if it
is not.
</p>
<p>If a second output argument is requested then the index into <var>s</var> of each
matching element is also returned.
</p>
<div class="example">
<pre class="example">a = [3, 10, 1];
s = [0:9];
[tf, s_idx] = ismember (a, s)
⇒ tf = [1, 0, 1]
⇒ s_idx = [4, 0, 2]
</pre></div>
<p>The inputs <var>a</var> and <var>s</var> may also be cell arrays.
</p>
<div class="example">
<pre class="example">a = {"abc"};
s = {"abc", "def"};
[tf, s_idx] = ismember (a, s)
⇒ tf = 1
⇒ s_idx = 1
</pre></div>
<p>If the optional third argument <code>"rows"</code> is given then compare rows
in <var>a</var> with rows in <var>s</var>. The inputs must be 2-D matrices with the
same number of columns to use this option.
</p>
<div class="example">
<pre class="example">a = [1:3; 5:7; 4:6];
s = [0:2; 1:3; 2:4; 3:5; 4:6];
[tf, s_idx] = ismember (a, s, "rows")
⇒ tf = logical ([1; 0; 1])
⇒ s_idx = [2; 0; 5];
</pre></div>
<p><strong>See also:</strong> <a href="Finding-Elements-and-Checking-Conditions.html#XREFlookup">lookup</a>, <a href="Sets.html#XREFunique">unique</a>, <a href="#XREFunion">union</a>, <a href="#XREFintersect">intersect</a>, <a href="#XREFsetdiff">setdiff</a>, <a href="#XREFsetxor">setxor</a>.
</p></dd></dl>
<span id="XREFpowerset"></span><dl class="def">
<dt id="index-powerset"><span class="category">: </span><span><em></em> <strong>powerset</strong> <em>(<var>a</var>)</em><a href='#index-powerset' class='copiable-anchor'> ¶</a></span></dt>
<dt id="index-powerset-1"><span class="category">: </span><span><em></em> <strong>powerset</strong> <em>(<var>a</var>, "rows")</em><a href='#index-powerset-1' class='copiable-anchor'> ¶</a></span></dt>
<dd><p>Compute the powerset (all subsets) of the set <var>a</var>.
</p>
<p>The set <var>a</var> must be a numerical matrix or a cell array of strings. The
output will always be a cell array of either vectors or strings.
</p>
<p>With the optional argument <code>"rows"</code>, each row of the set <var>a</var> is
considered one element of the set. The input must be a 2-D numeric matrix
to use this argument.
</p>
<p><strong>See also:</strong> <a href="Sets.html#XREFunique">unique</a>, <a href="#XREFunion">union</a>, <a href="#XREFintersect">intersect</a>, <a href="#XREFsetdiff">setdiff</a>, <a href="#XREFsetxor">setxor</a>, <a href="#XREFismember">ismember</a>.
</p></dd></dl>
</div>
<hr>
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