Answer to Question #148116 in Discrete Mathematics for Promise Omiponle

Question #148116
Let P1={B0, B1, B2} be a partition of Z, where B0={3n|n ∈ Z}, B1={3n+ 1|n ∈ Z}, and B2={3n+ 2|n ∈ Z}. Describe the equivalence relation R1 corresponding to P1.
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Expert's answer
2020-12-15T19:49:46-0500

By definition, "(a,b)\\in R_1" if and only if "a,b\\in B_k"  for some "k\\in\\{1,2,3\\}." The set "B_0=\\{3n\\ :\\ n \\in\\mathbb Z\\}" contains all integers "a"  that have 0 as the remainder of the Euclidean division of "a" by 3. The set "B_1=\\{3n+1\\ :\\ n \\in\\mathbb Z\\}"  contains all integers "a" that have 1 as the remainder of the Euclidean division of "a" by 3. And the set "B_2=\\{3n+2\\ :\\ n \\in\\mathbb Z\\}" contains all integers "a" that have 2 as the remainder of the Euclidean division of "a" by 3.

   

Therefore, "(a,b)\\in R_1" if and only if "a" and "b" have the same remainder of the Euclidean division by 3. 



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