Prove the following Proposition: On Z_n, both * and + are commutative and associative. The identity for Z_n with * is [1] and the identity for + is [0].Please submit response as either a PDF or MS Word file.Infinite thanks.See attachment.Dear student,Please, see the attachment. The proof is given in all details. Best wishes,

OTA Prove the following Proposition:

On Z_n, both ⨀ and ⊕ are commutative and associative.

The identity for Z_n with ⨀ is [1] and the identity for ⊕ is [0].Proof. Recall that Z_n is the set of residue classes modulo n. Also, recall that the addition and multiplication are defined on Z_n as follows. Let C and C be residue classes modulo n. Choose any representative c of C and c of C. Then C⊕C is defined as a residue class containing c⊕c, while C⨀C is defined as a residue class containing c⨀c. In other words,

C⊕C=[c+c],

and

C⨀C=[c∙c]. Here [c] denotes the residue class containing c.

Let us now show that the addition ⊕ is commutative. We have

C⊕C=[c+c]=[c+c]=[c] ⊕ [c]=C⊕C, for any residue classes C and C modulo n.

This implies that the addition is commutative.

Similarly,

C⨀C=[c∙c]=[c∙c]=[c] ⨀ [c]=C⨀C.

Therefore, the multiplication ⨀ is commutative too.

Let us now prove that [1] is the identity for the multiplication . To this end, consider any residue class C and its representative c. We have

C⨀ [1]=[c] ⨀ [1]=[c∙1]=[c]=C.

Thus [1] is indeed an identity for ⨀ .

Since we have

C⊕ [0]=[c] ⊕ [0]=[c+0]=[c],

We conclude that [0] is an identity for ⊕.

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