<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>Set ** Set -- Cartesian product</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Set_sp_pl_sp__Set.html">next</a> | <a href="___Set_sp_st_sp__Set.html">previous</a> | <a href="___Set_sp_pl_sp__Set.html">forward</a> | <a href="___Set_sp_st_sp__Set.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>Set ** Set -- Cartesian product</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>x ** y</tt></div>
</dd></dl>
</div>
</li>
<li><span>Operator: <a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>x</tt>, <span>a <a href="___Set.html">set</a></span></span></li>
<li><span><tt>y</tt>, <span>a <a href="___Set.html">set</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Set.html">set</a></span>, whose elements are the sequences (a,b), where a is an element of x, and b is an element of y.</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : set {1,2} ** set {a,b,c}
o1 = set {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
o1 : Set</pre>
</td></tr>
</table>
Suppose we wish to form the set of all triples with entries either in the set A below.<table class="examples"><tr><td><pre>i2 : A = set{1,2}
o2 = set {1, 2}
o2 : Set</pre>
</td></tr>
<tr><td><pre>i3 : A ** A ** A
o3 = set {((1, 1), 1), ((1, 1), 2), ((1, 2), 1), ((1, 2), 2), ((2, 1), 1),
------------------------------------------------------------------------
((2, 1), 2), ((2, 2), 1), ((2, 2), 2)}
o3 : Set</pre>
</td></tr>
</table>
To make this last a set of triples, <a href="_splice.html" title="remove subsequences">splice</a> each element together. Or, use <a href="___Tally_sp^_st_st_sp__Z__Z.html" title="Cartesian power of sets and tallies">Tally ^** ZZ</a>.<table class="examples"><tr><td><pre>i4 : (A ** A ** A)/splice
o4 = set {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2),
------------------------------------------------------------------------
(2, 2, 1), (2, 2, 2)}
o4 : Set</pre>
</td></tr>
<tr><td><pre>i5 : A^**3
o5 = set {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2),
------------------------------------------------------------------------
(2, 2, 1), (2, 2, 2)}
o5 : Set</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Set.html" title="the class of all sets">Set</a> -- the class of all sets</span></li>
</ul>
</div>
</div>
</body>
</html>
