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. logic1 http://www.openmath.org/cd http://www.openmath.org/cd/logic1.ocd 2006-03-30 2004-03-30 4 0 Author: OpenMath Consortium SourceURL: https://github.com/OpenMath/CDs official This CD holds the basic logic functions. equivalent application This symbol is used to show that two boolean expressions are logically equivalent, that is have the same boolean value for any inputs. The condition (A is equivalent to B) is equivalent to the condition that (A implies B and B implies A) not application This symbol represents the logical not function which takes one boolean argument, and returns the opposite boolean value. for all x | not(not(x))=x and application This symbol represents the logical and function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if all arguments are true or false otherwise. for all x | x and not(x) = false nand application This symbol represents the logical nand function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if all arguments are true or true otherwise. for all x | x nand not(x) = true xor application This symbol represents the logical xor function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if there are an odd number of true arguments or false otherwise. for all x | x xor x = false for all x | x xor not(x) = true xnor application This symbol represents the logical xnor function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if there are an odd number of true arguments or true otherwise. for all x | x xnor x = true for all x | x xnor not(x) = false or application This symbol represents the logical or function which is an n-ary function taking boolean arguments and returning a boolean value. It is true if any of the arguments are true or false otherwise. for all x | x or not(x) = true for all a,b | not(a and b)= (not(a) or not(b)) nor application This symbol represents the logical nor function which is an n-ary function taking boolean arguments and returning a boolean value. It is false if any of the arguments are true or true otherwise. for all x | x nor not(x) = false for all a,b | a and b = (not(a) nor not(b)) implies application This symbol represents the logical implies function which takes two boolean expressions as arguments. It evaluates to false if the first argument is true and the second argument is false, otherwise it evaluates to true. for all x | false implies x true constant This symbol represents the boolean value true. not true = false false constant This symbol represents the boolean value false. not false = true