Please answer letter D and E onlyyy.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please answer letter D and E onlyyy.
![1.
Consider the set Z, × Z3 = {(x, y) : x E Z, and y e Z3}. Define the
operation/sum O in Z2 × Z3 such that for any two elements (a, b) and (c, d)
in Z, x Z3, we have (a,b) O (c, d) = (a +2 c, b +3 d) where +2 is the
addition modulo 2 in Z2, and +2 is the addition modulo 3 in Z3.
a. List the element of Z, × Z3.
b. Determine the identity element in Z, × Z3.
c. Determine the inverse element of each element in Z, × Z3.
d. Is Z, x Z, a group under the given operation? Why? Is it Abelian?
e. List a proper subgroup of Z, × Zz. Write "none" if there are no proper
subgroups.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F618e560d-07ee-4542-9637-a52e3ff943a2%2F31cfc4ee-f1d7-46c8-b66a-53ae839f9c14%2F5v8x8h9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
Consider the set Z, × Z3 = {(x, y) : x E Z, and y e Z3}. Define the
operation/sum O in Z2 × Z3 such that for any two elements (a, b) and (c, d)
in Z, x Z3, we have (a,b) O (c, d) = (a +2 c, b +3 d) where +2 is the
addition modulo 2 in Z2, and +2 is the addition modulo 3 in Z3.
a. List the element of Z, × Z3.
b. Determine the identity element in Z, × Z3.
c. Determine the inverse element of each element in Z, × Z3.
d. Is Z, x Z, a group under the given operation? Why? Is it Abelian?
e. List a proper subgroup of Z, × Zz. Write "none" if there are no proper
subgroups.
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