Answer to Question #145983 in Abstract Algebra for Sourav Mondal

Question #145983
Define a relation R on the set of integers Z
by R= {(n, n+ 3k) I k belongs to Z}. Show that R is an equivalence relation. Also find all distinct
equivalence classes.
1
Expert's answer
2020-11-24T05:26:48-0500

By given relation, "aRb" if "b = a+3k, k \\in \\Z" .

Reflexive"a = a+3 (0)" "\\implies aRa" . So "R" is an reflexive relation.

Symmetric: Let "aRb \\implies b = a+3k" where "k" is an integer.

"\\implies a = b - 3k = b + 3(-k) = b + 3k_1" and "k_1 = -k \\in \\Z" "\\implies bRa".

So. "R" is an symmetric relation.

Transitive: Let "aRb, bRc" "\\implies b = a+3k_1 , c =b +3k_2"

"\\implies c = a+3k_1 +3k_2 = a+3(k_1+k_2) = a+3k" where "k = k_1+k_2 \\in \\Z".

Hence, "aRc" . So, "R" is an transitive relation.

Thus, "R" is an equivalence relation.


Equivalence class of 0 = "\\{3k: k\\in \\Z\\} = \\{ \\_ \\ \\_ \\ \\_, -6, -3, 0, 3 , 6, \\_ \\ \\_ \\ \\_ \\}"

Equivalence class of 1 = "\\{3k+1 : k\\in \\Z\\} = \\{ \\_ \\ \\_ \\ \\_, -5, -2, 1 , 4 , 7, \\_ \\ \\_ \\ \\_ \\}"

Equivalence class of 2 = "\\{3k+2 : k\\in \\Z\\} = \\{ \\_ \\ \\_ \\ \\_, -4, -1, 2 , 5 , 8, \\_ \\ \\_ \\ \\_ \\}"

These three distinct equivalence classes are possible.



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