chap34defs - Part A. Equivalence Relations. Definition. Let...

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Part A. Equivalence Relations. 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. A binary relation R A × A is an equivalence relation if R is reflexive, symmetric, and transitive. Definition. Let m 2 be an integer. Define a binary relation called modulus m and denoted m , over the set of integers Z , as follows: for integers a and b , a m b iff m | ( a b ), that is, a b = mq for some integer q . Two numbers a and b such that a m b , are called congruent mod m , often denoted a b (mod m ). Theorem.
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chap34defs - Part A. Equivalence Relations. Definition. Let...

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