Question #122250
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
2020-06-15T19:19:11-0400

By given relation, aRb if b=a+3k,k∈Z .

Reflexive: a=a+3(0) ⟹ aRa . So R is an reflexive relation.

Symmetric: Let aRbb=a+3k where k is an integer.

    a=b3k=b+3(k)=b+3k1\implies a = b - 3k = b + 3(-k) = b + 3k_1

​ and k1=kZ    bRak_1 = -k \in \Z \implies bRa .

So. R is an symmetric relation.

Transitive: Let aRb,bRc     b=a+3k1,c=b+3k2\implies b = a+3k_1 , c =b +3k_2

    c=a+3k1+3k2=a+3(k1+k2)=a+3k\implies c = a+3k_1 +3k_2 = a+3(k_1+k_2) = a+3k

where k=k1+k2Z.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:kZ}={_ _ _,6,3,0,3,6,_ _ _}= \{3k: k\in \Z\} = \{ \_ \ \_ \ \_, -6, -3, 0, 3 , 6, \_ \ \_ \ \_ \}

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

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

These three distinct equivalence classes are possible


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