Let n be a natural number n greater than equals to 2 Define

Let n be a natural number n greater than equals to 2. Define [a] = {b Belongs to Z : a b(mod n)} and Z_n = {[x] : x belongs to Z}. As mentioned in class : is an equivalence relation on the integers. Prove that Z_n = {[0] , [1],....,[n-1]}. Use the division algorithm and the fact that equivalence Define [a]+[b] = [a+b]. If [x] = [y],[z] = [w], Prove that [x]+[z] = [y]+[w]. Show that Z_n is a group under the operation defined in part(b).

Solution

a can take values as 0,1...n-1 only

These form a group since 0 is identity element, and closed under addition.

Any two numbers x+y is less than n will be in Zn or if > n, then x+y mod n will be in Zn.

Inverse of any element x = n-x is in Zn for all x in Zn

Hence Zn is a group

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