Let I be the set of equivalence classes as defined as above. Define O as follows: if A and B are equivalence classes in T, A O B is the equivalence class of the sum of an element from A and an element from B. For example, using the A and B from #6: (2,5) + (4,2) = (6,7), so, A Ð B is the equivalence class of (6,7): {(0,1), (1,2), (2,3), ...} Đis well defined because as we showed in #6, the result of the operation does not depend on which representatives of the equivalence classes we choose.
Let I be the set of equivalence classes as defined as above. Define O as follows: if A and B are equivalence classes in T, A O B is the equivalence class of the sum of an element from A and an element from B. For example, using the A and B from #6: (2,5) + (4,2) = (6,7), so, A Ð B is the equivalence class of (6,7): {(0,1), (1,2), (2,3), ...} Đis well defined because as we showed in #6, the result of the operation does not depend on which representatives of the equivalence classes we choose.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...
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