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 ⊕.




Why Choose Us

  • 100% non-plagiarized Papers
  • 24/7 /365 Service Available
  • Affordable Prices
  • Any Paper, Urgency, and Subject
  • Will complete your papers in 6 hours
  • On-time Delivery
  • Money-back and Privacy guarantees
  • Unlimited Amendments upon request
  • Satisfaction guarantee

How it Works

  • Click on the “Place Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
  • Fill in your paper’s requirements in the "PAPER DETAILS" section.
  • Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
  • Click “CREATE ACCOUNT & SIGN IN” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page.
  • From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.