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<html>
<head>
<title>
SET_THEORY - An Implementation of Set Theoretic Operations
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SET_THEORY <br> An Implementation of Set Theoretic Operations
</h1>
<hr>
<p>
<b>SET_THEORY</b>
is a C++ library which
implements some of the operations of set theory.
</p>
<p>
We assume that a set is represented by a strictly ascending sequence
of positive integers. We might think of a universal set U = 1 : N
in cases where all our subsets will have elements between 1 and N.
</p>
<p>
Set theoretic operations include:
<ul>
<li>
<i>definition</i> using a numeric property: A = <b>find</b> ( mod ( U, 3 ) = 1 );
</li>
<li>
<i>definition</i> using an explicit list: A = <b>[</b> 1, 3, 8<b>]</b>;
</li>
<li>
<i>unique</i>: creating a set from the unique elements of a list:
A = <b>unique</b> ( [ 1, 3, 8, 3, 3, 1, 7, 3, 1, 1 ] );
</li>
<li>
<i>union</i>: C = <b>union</b> ( A, B );
</li>
<li>
<i>intersection</i>: C = <b>intersect</b> ( A, B );
</li>
<li>
<i>symmetric difference</i>: C = <b>setxor</b> ( A, B );
</li>
<li>
<i>complement</i> with respect to the universal set: B = <b>setdiff</b> ( U, A );
</li>
<li>
<i>complement</i> with respect to another set: C = <b>setdiff</b> ( B, A );
</li>
<li>
<i>cardinality</i>: n = <b>length</b> ( A );
</li>
<li>
<i>membership</i>: true/false = <b>ismember</b> ( a, A );
</li>
<li>
<i>subset</i>: true/false = <b>ismember</b> ( B, A );
</li>
<li>
<i>addition of one element</i>: A = <b>unique</b> ( [ A; a ] );
</li>
<li>
<i>deletion of one element</i>: A = <b>setxor</b> ( A, a );
</li>
<li>
<i>indexing one element</i>: i = <b>find</b> ( A == a );
</li>
</ul>
</p>
<p>
Although sets are traditionally allowed to contain arbitrary elements,
it is computationally convenient to assume that our sets are simply
subsets of the integers from 1 to N.
</p>
<p>
If N is no greater than 32, we can represent a set using a
32 bit integer. We term this the <b>B4SET</b> representation.
It is compact, but as it stands, is limited to a universal
set of no more than 32 elements.
</p>
<p>
Assuming we can regard the integer as an unsigned
quantity, each bit of the binary representation of the integer
represents the presence (1) or absence (0) of the corresponding
integer in the set. Thus, assuming N is 5, the set { 1, 2, 5}
corresponds to the binary representation 10011 and hence to the
integer 19. In order to read or write the individual bits of
an integer, the functions BTEST, IBCLR and IBSET are useful in
this case.
</p>
<p>
A more flexible, but less efficient, representation of sets
uses a logical vector, and is called the <b>LSET</b> representation.
Assuming we have a universal set of N elements, any set is represented
by a logical vector of N elements, the I-th element of which is
TRUE if I is an element of the set.
</p>
<p>
A representation that can be more efficient for small subsets of
a large universal set is the <b>I4SET</b>. In this representation,
we simply list, in ascending order, the elements of the set.
The representation is simple, but manipulation is more involved.
For instance, to create the union of two sets, we must determine
the number of unique elements in the two component sets, allocate
the necessary space, then interleave the elements of the two
components appropriately.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this
web page are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>SET_THEORY</b> is available in
<a href = "../../c_src/set_theory/set_theory.html">a C version</a> and
<a href = "../../cpp_src/set_theory/set_theory.html">a C++ version</a> and
<a href = "../../f77_src/set_theory/set_theory.html">a FORTRAN77 version</a> and
<a href = "../../f_src/set_theory/set_theory.html">a FORTRAN90 version</a> and
<a href = "../../m_src/set_theory/set_theory.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/combo/combo.html">
COMBO</a>,
a C++ library which
handles combinatorial problems, by Kreher and Stinson;
</p>
<p>
<a href = "../../cpp_src/subset/subset.html">
SUBSET</a>,
a C++ library which
ranks, unranks, and generates random subsets, combinations, permutations, and so on;
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Charles Pinter,<br>
Set Theory,<br>
Addison-Wesley, 1971,<br>
LC: QA248.P55.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "set_theory.cpp">set_theory.cpp</a>, the source code.
</li>
<li>
<a href = "set_theory.hpp">set_theory.hpp</a>, the include file.
</li>
<li>
<a href = "set_theory.sh">set_theory.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "set_theory_prb.cpp">set_theory_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "set_theory_prb.sh">set_theory_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "set_theory_prb_output.txt">set_theory_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>B4SET_COLEX_RANK</b> computes the colexicographic rank of a B4SET.
</li>
<li>
<b>B4SET_COLEX_SUCCESSOR</b> computes the colexicographic successor of a B4SET.
</li>
<li>
<b>B4SET_COLEX_UNRANK</b> computes the B4SET of given colexicographic rank.
</li>
<li>
<b>B4SET_COMPLEMENT</b> computes the complement of a B4SET.
</li>
<li>
<b>B4SET_COMPLEMENT_RELATIVE</b> computes the relative complement of a B4SET.
</li>
<li>
<b>B4SET_DELETE</b> deletes an element from a B4SET.
</li>
<li>
<b>B4SET_DISTANCE</b> computes the Hamming distance between two B4SET's.
</li>
<li>
<b>B4SET_ENUM</b> enumerates the B4SET's.
</li>
<li>
<b>B4SET_INDEX</b> returns the index of an element of a B4SET.
</li>
<li>
<b>B4SET_INSERT</b> inserts an item into a B4SET.
</li>
<li>
<b>B4SET_INTERSECT</b> computes the intersection of two B4SET's.
</li>
<li>
<b>B4SET_IS_EMPTY</b> determines if a B4SET is empty.
</li>
<li>
<b>B4SET_IS_EQUAL</b> determines if two B4SET's are equal.
</li>
<li>
<b>B4SET_IS_MEMBER</b> determines if an item is a member of a B4SET.
</li>
<li>
<b>B4SET_IS_SUBSET</b> determines if one B4SET is a subset of another.
</li>
<li>
<b>B4SET_LEX_RANK</b> computes the lexicographic rank of a B4SET.
</li>
<li>
<b>B4SET_LEX_SUCCESSOR</b> computes the lexicographic successor of a B4SET.
</li>
<li>
<b>B4SET_LEX_UNRANK</b> computes the B4SET of given lexicographic rank.
</li>
<li>
<b>B4SET_RANDOM</b> sets a rondom B4SET.
</li>
<li>
<b>B4SET_TO_LSET</b> converts a B4SET to an LSET.
</li>
<li>
<b>B4SET_TRANSPOSE_PRINT</b> prints a B4SET "transposed".
</li>
<li>
<b>B4SET_UNION</b> computes the union of two B4SET's.
</li>
<li>
<b>B4SET_WEIGHT</b> computes the Hamming weight of a B4SET.
</li>
<li>
<b>B4SET_XOR</b> computes the symmetric difference of two B4SET's.
</li>
<li>
<b>DIGIT_TO_CH</b> returns the base 10 digit character corresponding to a digit.
</li>
<li>
<b>I4_BCLR</b> clears a bit of an I4.
</li>
<li>
<b>I4_BSET</b> sets a bit of an I4.
</li>
<li>
<b>I4_BTEST</b> returns a bit of an I4.
</li>
<li>
<b>I4_LOG_10</b> returns the integer part of the logarithm base 10 of an I4.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum of two I4's.
</li>
<li>
<b>I4_POWER</b> returns the value of I^J.
</li>
<li>
<b>I4_TO_S</b> converts an I4 to a string.
</li>
<li>
<b>I4_WIDTH</b> returns the "width" of an I4.
</li>
<li>
<b>I4_XOR</b> xor's a bit of an I4.
</li>
<li>
<b>I4VEC_TO_B4SET</b> converts an I4VEC to a B4SET.
</li>
<li>
<b>I4VEC_TO_LSET</b> converts an I4VEC to an LSET.
</li>
<li>
<b>I4VEC_UNIFORM_NEW</b> returns a scaled pseudorandom I4VEC.
</li>
<li>
<b>LSET_COLEX_RANK</b> computes the colexicographic rank of an LSET.
</li>
<li>
<b>LSET_COLEX_SUCCESSOR</b> computes the colexicographic successor of an LSET.
</li>
<li>
<b>LSET_COLEX_UNRANK</b> computes the LSET of given colexicographic rank.
</li>
<li>
<b>LSET_COMPLEMENT</b> computes the complement of an LSET.
</li>
<li>
<b>LSET_COMPLEMENT_RELATIVE</b> computes the relative complement of an LSET.
</li>
<li>
<b>LSET_DELETE</b> deletes an element from an LSET.
</li>
<li>
<b>LSET_DISTANCE</b> computes the Hamming distance between two LSET's.
</li>
<li>
<b>LSET_ENUM</b> enumerates the LSET's.
</li>
<li>
<b>LSET_INDEX</b> returns the index of an element of an LSET.
</li>
<li>
<b>LSET_INSERT</b> inserts an item into an LSET.
</li>
<li>
<b>LSET_INTERSECT</b> computes the intersection of two LSET's.
</li>
<li>
<b>LSET_IS_EMPTY</b> determines if an LSET is empty.
</li>
<li>
<b>LSET_IS_EQUAL</b> determines if two LSET's are equal.
</li>
<li>
<b>LSET_IS_MEMBER</b> determines if an item is a member of an LSET.
</li>
<li>
<b>LSET_IS_SUBSET</b> determines if one LSET is a subset of another.
</li>
<li>
<b>LSET_LEX_RANK</b> computes the lexicographic rank of an LSET.
</li>
<li>
<b>LSET_LEX_SUCCESSOR</b> computes the lexicographic successor of an LSET.
</li>
<li>
<b>LSET_LEX_UNRANK</b> computes the LSET of given lexicographic rank.
</li>
<li>
<b>LSET_RANDOM</b> sets a rondom LSET.
</li>
<li>
<b>LSET_TO_B4SET</b> converts an I4VEC to a B4SET.
</li>
<li>
<b>LSET_TRANSPOSE_PRINT</b> prints an LSET "transposed".
</li>
<li>
<b>LSET_UNION</b> computes the union of two LSET's.
</li>
<li>
<b>LSET_WEIGHT</b> computes the Hamming weight of an LSET.
</li>
<li>
<b>LSET_XOR</b> computes the symmetric difference of two LSET's.
</li>
<li>
<b>LVEC_TRANSPOSE_PRINT</b> prints an LVEC "transposed".
</li>
<li>
<b>R4_NINT</b> returns the nearest integer to an R4.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 20 September 2011.
</i>
<!-- John Burkardt -->
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</html>