OpenMath Content Dictionary: setname1

Canonical URL:
http://www.openmath.org/cd/setname1.ocd
CD File:
setname1.ocd
CD as XML Encoded OpenMath:
setname1.omcd
Defines:
C, N, P, Q, R, Z
Date:
2001-03-12
Version:
2
Review Date:
2003-04-01
Status:
official
Uses CD:
alg1, arith1, logic1, quant1, relation1, set1, nums1 

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

This CD defines common sets of mathematics 

Written by J.H. Davenport on 1999-04-18.
Revised to add Zm, GFp, GFpn on 1999-11-09.
Revised to add QuotientField and A on 1999-11-19.

P

This symbol represents the set of positive prime numbers.

Commented Mathematical property (CMP):
for all n | n is a positive prime number is equivalent to: n is a natural number and n > 1 and ((n=a*b and a and b are natural numbers) implies ((a=1 and b=n) or (b=1 and a=n)))
Formal Mathematical property (FMP): 
<OMOBJ>
  <OMBIND>
    <OMS name="forall" cd="quant1"/>
    <OMBVAR>
       <OMV name="n"/>
    </OMBVAR>
    <OMA>
      <OMS name="equivalent" cd="logic1"/>
      <OMA>
        <OMS name="in" cd="set1"/>
        <OMV name="n"/>
        <OMS name="P" cd="setname1"/>
      </OMA>
      <OMA>
        <OMS name="and" cd="logic1"/>
        <OMA>
          <OMS name="in" cd="set1"/>
          <OMV name="n"/>
          <OMS name="N" cd="setname1"/>
        </OMA>
        <OMA>
          <OMS name="gt" cd="relation1"/>
          <OMV name="n"/>
          <OMS name="one" cd="alg1"/>
        </OMA>
        <OMA>
          <OMS name="implies" cd="logic1"/>
          <OMA>
            <OMS name="and" cd="logic1"/>
            <OMA>
              <OMS name="eq" cd="relation1"/>
              <OMV name="n"/>
              <OMA>
                <OMS name="times" cd="arith1"/>
                <OMV name="a"/>
                <OMV name="b"/>
              </OMA>
            </OMA>
            <OMA>
              <OMS name="in" cd="set1"/>
              <OMV name="a"/>
              <OMS name="N" cd="setname1"/>
            </OMA>
            <OMA>
              <OMS name="in" cd="set1"/>
              <OMV name="b"/>
              <OMS name="N" cd="setname1"/>
            </OMA>
          </OMA>
          <OMA>
            <OMS name="or" cd="logic1"/>
            <OMA>
              <OMS name="and" cd="logic1"/>
              <OMA>
                <OMS name="eq" cd="relation1"/>
                <OMV name="a"/>
                <OMS name="one" cd="alg1"/>
              </OMA>
              <OMA>
                <OMS name="eq" cd="relation1"/>
                <OMV name="b"/>
                <OMV name="n"/>
              </OMA>
            </OMA>
            <OMA>
              <OMS name="and" cd="logic1"/>
              <OMA>
                <OMS name="eq" cd="relation1"/>
                <OMV name="b"/>
                <OMS name="one" cd="alg1"/>
              </OMA>
              <OMA>
                <OMS name="eq" cd="relation1"/>
                <OMV name="a"/>
                <OMV name="n"/>
              </OMA>
            </OMA>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>

forall [ n ] . (equivalent (in ( n, P) , and (in ( n, N) , gt ( n, one) , implies (and (eq ( n, times ( a, b) ) , in ( a, N) , in ( b, N) ) , or (and (eq ( a, one) , eq ( b, n) ) , and (eq ( b, one) , eq ( a, n) ) ) ) ) ) )

Signatures:
sts

[Next: N] [Last: C] [Top]

N

This symbol represents the set of natural numbers (including zero).

Commented Mathematical property (CMP):
for all n | n in the natural numbers is equivalent to saying n=0 or n-1 is a natural number
Formal Mathematical property (FMP): 
<OMOBJ>
  <OMBIND>
    <OMS name="forall" cd="quant1"/>
    <OMBVAR>
       <OMV name="n"/>
    </OMBVAR>
    <OMA>
      <OMS name="implies" cd="logic1"/>
      <OMA>
        <OMS name="in" cd="set1"/>
        <OMV name="n"/>
        <OMS name="N" cd="setname1"/>
      </OMA>
      <OMA>
        <OMS name="or" cd="logic1"/>
        <OMA>
          <OMS name="eq" cd="relation1"/>
          <OMV name="n"/>
          <OMS name="zero" cd="alg1"/>
        </OMA>
        <OMA>
          <OMS name="in" cd="set1"/>
          <OMA>
            <OMS name="minus" cd="arith1"/>
            <OMV name="n"/>
            <OMS name="one" cd="alg1"/>
          </OMA>
          <OMS name="N" cd="setname1"/>
        </OMA>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>

forall [ n ] . (implies (in ( n, N) , or (eq ( n, zero) , in (minus ( n, one) , N) ) ) )

Signatures:
sts

[Next: Z] [Previous: P] [Top]

Z

This symbol represents the set of integers, positive, negative and zero.

Commented Mathematical property (CMP):
for all z | the statements z is an integer and z is a natural number or -z is a natural number are equivalent
Formal Mathematical property (FMP): 
<OMOBJ>
  <OMBIND>
    <OMS name="forall" cd="quant1"/>
    <OMBVAR>
       <OMV name="z"/>
    </OMBVAR>
    <OMA>
      <OMS name="implies" cd="logic1"/>
      <OMA>
        <OMS name="in" cd="set1"/>
        <OMV name="z"/>
        <OMS name="Z" cd="setname1"/>
      </OMA>
      <OMA>
        <OMS name="or" cd="logic1"/>
        <OMA>
          <OMS name="in" cd="set1"/>
          <OMV name="z"/>
          <OMS name="N" cd="setname1"/>
        </OMA>
        <OMA>
          <OMS name="in" cd="set1"/>
          <OMA>
            <OMS name="unary_minus" cd="arith1"/>
            <OMV name="z"/>
          </OMA>
          <OMS name="N" cd="setname1"/>
        </OMA>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>

forall [ z ] . (implies (in ( z, Z) , or (in ( z, N) , in (unary_minus ( z) , N) ) ) )

Signatures:
sts

[Next: Q] [Previous: N] [Top]

Q

This symbol represents the set of rational numbers.

Commented Mathematical property (CMP):
for all z where z is a rational, there exists integers p and q with q > 1 and p/q = z
Formal Mathematical property (FMP): 
<OMOBJ>
  <OMBIND>
    <OMS name="forall" cd="quant1"/>
    <OMBVAR>
       <OMV name="z"/>
    </OMBVAR>
    <OMA>
      <OMS name="implies" cd="logic1"/>
      <OMA>
        <OMS name="in" cd="set1"/>
        <OMV name="z"/>
        <OMS name="Q" cd="setname1"/>
      </OMA>
      <OMBIND>
        <OMS name="exists" cd="quant1"/>
        <OMBVAR>
          <OMV name="p"/>
          <OMV name="q"/>
        </OMBVAR>
        <OMA>
          <OMS name="and" cd="logic1"/>
          <OMA>
            <OMS name="in" cd="set1"/>
            <OMV name="p"/>
            <OMS name="Z" cd="setname1"/>
          </OMA>
          <OMA>
            <OMS name="in" cd="set1"/>
            <OMV name="q"/>
            <OMS name="Z" cd="setname1"/>
          </OMA>
          <OMA>
            <OMS name="geq" cd="relation1"/>
            <OMV name="q"/>
            <OMS name="one" cd="alg1"/>
          </OMA>
          <OMA>
            <OMS name="eq" cd="relation1"/>
            <OMV name="z"/>
            <OMA>
              <OMS name="divide" cd="arith1"/>
              <OMV name="p"/>
              <OMV name="q"/>
            </OMA>
          </OMA>
        </OMA>
      </OMBIND>
     </OMA>
  </OMBIND>
</OMOBJ>

forall [ z ] . (implies (in ( z, Q) , exists [ p q ] . (and (in ( p, Z) , in ( q, Z) , geq ( q, one) , eq ( z, divide ( p, q) ) ) ) ) )

Commented Mathematical property (CMP):
for all a,b | a,b rational with a<b implies there exists rational a,c s.t. a<c and c<b
Formal Mathematical property (FMP): 
<OMOBJ>
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="a"/>
    <OMV name="b"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="in"/>
	<OMV name="a"/>
	<OMS cd="setname1" name="Q"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="in"/>
	<OMV name="b"/>
	<OMS cd="setname1" name="Q"/>
      </OMA>
      <OMA>
        <OMS cd="relation1" name="lt"/>
	<OMV name="a"/>
	<OMV name="b"/>
      </OMA>
    </OMA>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
        <OMV name="c"/>
      </OMBVAR>
      <OMA>
        <OMS cd="logic1" name="and"/>
	<OMA>
	  <OMS cd="set1" name="in"/>
	  <OMV name="c"/>
	  <OMS cd="setname1" name="Q"/>
	</OMA>
	<OMA>
	  <OMS cd="relation1" name="lt"/>
	  <OMV name="a"/>
	  <OMV name="c"/>
	</OMA>
	<OMA>
	  <OMS cd="relation1" name="lt"/>
	  <OMV name="c"/>
	  <OMV name="b"/>
	</OMA>
      </OMA>
    </OMBIND>
  </OMA>
</OMBIND>
</OMOBJ>

forall [ a b ] . (implies (and (in ( a, Q) , in ( b, Q) , lt ( a, b) ) , exists [ c ] . (and (in ( c, Q) , lt ( a, c) , lt ( c, b) ) ) ) )

Signatures:
sts

[Next: R] [Previous: Z] [Top]

R

This symbol represents the set of real numbers.

Commented Mathematical property (CMP):
S \subset R and exists y in R : forall x in S x <= y) implies exists z in R such that (( forall x in S x <= z) and ((forall x in S x <= w) implies z <= w)
Formal Mathematical property (FMP): 
<OMOBJ>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="subset"/>
	<OMV name="S"/>
	<OMS cd="setname1" name="R"/>
      </OMA>
      <OMBIND>
        <OMS cd="quant1" name="exists"/>
	<OMBVAR>
	  <OMV name="y"/>
	</OMBVAR>
	<OMA>
	  <OMS cd="logic1" name="and"/>
	  <OMA>
	    <OMS cd="set1" name="in"/>
	    <OMV name="y"/>
	    <OMS cd="setname1" name="R"/>
	  </OMA>
	  <OMBIND>
	    <OMS cd="quant1" name="forall"/>
	    <OMBVAR>
	      <OMV name="x"/>
	    </OMBVAR>
	    <OMA>
	      <OMS cd="logic1" name="and"/>
	      <OMA>
	        <OMS cd="set1" name="in"/>
		<OMV name="x"/>
		<OMV name="S"/>
	      </OMA>
	      <OMA>
	        <OMS cd="relation1" name="leq"/>
		<OMV name="x"/>
		<OMV name="y"/>
	      </OMA>
	    </OMA>
	  </OMBIND>
	</OMA>
      </OMBIND>
    </OMA>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
        <OMV name="z"/>
      </OMBVAR>
      <OMA>
        <OMS cd="logic1" name="and"/>
	<OMA>
          <OMS cd="set1" name="in"/>
	  <OMV name="z"/>
	  <OMS cd="setname1" name="R"/>
	</OMA>
	<OMBIND>
	  <OMS cd="quant1" name="forall"/>
	  <OMBVAR>
	    <OMV name="x"/>
	  </OMBVAR>
	  <OMA>
	    <OMS cd="logic1" name="implies"/>
	    <OMA>
	      <OMS cd="set1" name="in"/>
	      <OMV name="x"/>
	      <OMV name="S"/>
	    </OMA>
	    <OMA>
	      <OMS cd="relation1" name="leq"/>
	      <OMV name="x"/>
	      <OMV name="z"/>
	    </OMA>
	  </OMA>
	</OMBIND>
	<OMA>
	  <OMS cd="logic1" name="implies"/>
	  <OMBIND>
	    <OMS cd="quant1" name="forall"/>
	    <OMBVAR>
	      <OMV name="x"/>
	    </OMBVAR>
	    <OMA>
	      <OMS cd="logic1" name="implies"/>
	      <OMA>
	        <OMS cd="set1" name="in"/>
		<OMV name="x"/>
		<OMV name="S"/>
	      </OMA>
	      <OMA>
	        <OMS cd="relation1" name="leq"/>
		<OMV name="x"/>
		<OMV name="w"/>
	      </OMA>
	    </OMA>
	  </OMBIND>
	  <OMA>
	    <OMS cd="relation1" name="leq"/>
	    <OMV name="z"/>
	    <OMV name="w"/>
	  </OMA>
	</OMA>
      </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>

implies (and (subset ( S, R) , exists [ y ] . (and (in ( y, R) , forall [ x ] . (and (in ( x, S) , leq ( x, y) ) ) ) ) ) , exists [ z ] . (and (in ( z, R) , forall [ x ] . (implies (in ( x, S) , leq ( x, z) ) ) , implies (forall [ x ] . (implies (in ( x, S) , leq ( x, w) ) ) , leq ( z, w) ) ) ) )

Signatures:
sts

[Next: C] [Previous: Q] [Top]

C

This symbol represents the set of complex numbers.

Commented Mathematical property (CMP):
for all z | if z is complex then there exist reals x,y s.t. z = x + i * y
Formal Mathematical property (FMP): 
<OMOBJ>
<OMBIND>
  <OMS cd="quant1" name="forall"/>
  <OMBVAR>
    <OMV name="z"/>
  </OMBVAR>
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="z"/>
      <OMS cd="setname1" name="C"/>
    </OMA>
    <OMBIND>
      <OMS cd="quant1" name="exists"/>
      <OMBVAR>
        <OMV name="x"/>
	<OMV name="y"/>
      </OMBVAR>
      <OMA>
        <OMS cd="logic1" name="and"/>
	<OMA>
	  <OMS cd="set1" name="in"/>
	  <OMV name="x"/>
	  <OMS cd="setname1" name="R"/>
	</OMA>
	<OMA>
	  <OMS cd="set1" name="in"/>
	  <OMV name="y"/>
	  <OMS cd="setname1" name="R"/>
	</OMA>
	<OMA>
	  <OMS cd="relation1" name="eq"/>
	  <OMV name="z"/>
	  <OMA>
	    <OMS cd="arith1" name="plus"/>
	    <OMV name="x"/>
	    <OMA>
	      <OMS cd="arith1" name="times"/>
	      <OMS cd="nums1" name="i"/>
	      <OMV name="y"/>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
    </OMBIND>
  </OMA>
</OMBIND>
</OMOBJ>

forall [ z ] . (implies (in ( z, C) , exists [ x y ] . (and (in ( x, R) , in ( y, R) , eq ( z, plus ( x, times (i, y) ) ) ) ) ) )

Signatures:
sts

[First: P] [Previous: R] [Top]