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# Relations in a cross-compiler

7/24/2010 8:00:26 PM
##### 1.5 RELATIONS Let A and B be the two sets; then the relationship R between A and B is nothing more than a set of ordered pairs (a, b) such that a is in A and b is in B, and a is related to b by relation R. That is: R = { (a, b) | a is in A and b is in B, and a is related to b by R } For example, if A = { 0, 1 } and B = { 1, 2 }, then we can define a relation of ‘less than,’ denoted by < as follows: A pair (1, 1) will not belong to the < relation, because one is not less than one. Therefore, we conclude that a relation R between sets A and B is the subset of A � B. If a pair (a, b) is in R, then aRb is true; otherwise, aRb is false. A is called a "domain" of the relation, and B is called a "range" of the relation. If the domain of a relation R is a set A, and the range is also a set A, then R is called as a relation on set A rather than calling a relation between sets A and B. For example, if A = { 0, 1, 2 }, then a < relation defined on A will result in: < = { (0, 1), (0, 2), (1, 2) }. ### Properties of the Relation Let R be some relation defined on a set A. Therefore: R is said to be reflexive if aRa is true for every a in A; that is, if every element of A is related with itself by relation R, then R is called as a reflexive relation. If every aRb implies bRa (i.e., when a is related to b by R, and if b is also related to a by the same relation R), then a relation R will be a symmetric relation. If every aRb and bRc implies aRc, then the relation R is said to be transitive; that is, when a is related to b by R, and b is related to c by R, and if a is also related to c by relation R, then R is a transitive relation. If R is reflexive and transitive, as well as symmetric, then R is an equivalence relation. #### Property Closure of a Relation Let R be a relation defined on a set A, and if P is a set of properties, then the property closure of a relation R, denoted as P-closure, is the smallest relation, R′, which has the properties mentioned in P. It is obtained by adding every pair (a, b) in R to R′, and then adding those pairs of the members of A that will make relation R have the properties in P. If P contains only transitivity properties, then the P-closure will be called as a transitive closure of the relation, and we denote the transitive closure of relation R by R+; whereas when P contains transitive as well as reflexive properties, then the P-closure is called as a reflexive-transitive closure of relation R, and we denote it by R*. R+ can be obtained from R as follows: For example, if: var sc_project=11388663; var sc_invisible=1; var sc_security="7db37af3"; var scJsHost = (("https:" == document.location.protocol) ? "https://secure." : "http://www."); document.write("<sc"+"ript type='text/javascript' src='" + scJsHost+ "statcounter.com/counter/counter.js'></"+"script>"); Other

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