chap2defs - Part A. Relations. Definition. A relation R...

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Part A. Relations. Definition. A relation R defined over sets A and B is a (i.e. any) subset of A × B , that is, R A × B . Such a relation is a binary relation; the degree of R is 2. Definition. An n -ary relation R over sets A 1 , A 2 , …, A n , n 2, is a subset of the cartesian product The degree of R is n . Definition. Let R A × B and S B × C denote two relations. The composition of R and S , denoted R S , is a binary relation over A and C defined as follows: R S = {( a , c )| a A and c C , and there exists b B such that aRb and bSc }. Theorem. Let R A × B , S B × C , and T C × D denote three binary relations. Then relation compositions satisfy the following associative law: ( R S ) T = R ( S T ). Theorem. Let R A × B , S B × C , and T B × C denote 3 binary relations. Then (1) R ( S T ) = ( R S ) ( R T ); (2) R ( S T ) ( R S ) ( R T ). (In general, the two sides of (2) are not equal.) Definition. Let R A × A be a binary relation. (1) R is reflexive if for all a A , aRa , i,e., ( a , a ) R . (In words, each element a of A is related to itself via R .) (2) R is irreflexive if for all a A , ( a , a ) R . (In words, each element a of A is not related to itself via R .) (3) R is symmetric if aRb implies bRa , i.e., for all ( a , b ) R , ( b , a ) R . (In words, whenever a is related to b , b is related to a .) (4) R is anti - symmetric if aRb and bRa imply a = b . Equivalently, this means if a b , then ( a , b ) R implies ( b , a ) R . (In words, whenever a is related to b , and if a b , then b is not related to a .) (5) R is transitive if aRb and bRc imply aRc , i.e., if ( a , b ) R and ( b , c ) R , then ( a , c ) R . (In words, whenever a is related to b , and b is related to c , then a is related to c .) Definition. Consider a binary relation R A × B . The inverse of R , denoted R –1 , is a binary relation
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chap2defs - Part A. Relations. Definition. A relation R...

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