Answer to Question #145983 in Abstract Algebra for Sourav Mondal

Question

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.

Answer 1

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.