Question 3 Take a look at the operator *, defined as follows: *: ZxZ→Z x*y = x + 3y (a.) Is the operator* commutative? Explain. (b.) Is the operator associative? Explain. (c.) Does the operator * have a unit element e ("identitet" in Swedish)? Explain (d.) Can you find two integers a and b that commute ("kommuterar") for *? Explain.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question
Question 3
Take a look at the operator *, defined as follows:
*: ZxZ→ Z
x+y = x + 3y
(a.) Is the operator ★ commutative? Explain.
(b.) Is the operator associative? Explain.
(c.) Does the operator have a unit element e ("identitet" in Swedish)? Explain.
(d.) Can you find two integers a and b that commute ("kommuterar") for *? Explain.
(Read “Lemurell and Johansson” if you need to know what associative, commutative, unit element is.
Transcribed Image Text:Question 3 Take a look at the operator *, defined as follows: *: ZxZ→ Z x+y = x + 3y (a.) Is the operator ★ commutative? Explain. (b.) Is the operator associative? Explain. (c.) Does the operator have a unit element e ("identitet" in Swedish)? Explain. (d.) Can you find two integers a and b that commute ("kommuterar") for *? Explain. (Read “Lemurell and Johansson” if you need to know what associative, commutative, unit element is.
Expert Solution
steps

Step by step

Solved in 3 steps with 1 images

Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

My teacher gave me a feedback, can you help me with it?
2. Almost! But there's nothing that maps to 0. In your notes at the top you've listed ℕ={1,2,3,…}, so you should keep in mind that (at least in computer science-adjacent fields) ℕ includes 0.

3c. It is true that an identity needs to be unique (as in there cannot exist more than one, which can be a fun exercise to prove if you feel so inclined), but I'm guessing you mean either that the left e=-2x is not constant, or that left e ≠ right e. Both are indeed necessary for it to be an identity, but the phrasing is unclear.

3d. That the operator is not commutative doesn't mean there are no particular pairs that commute — for example 2⁴=4²=16. In other words you have to show that x⭑y=y⭑x if *and only if* x=y. To prove this absence, it might be be helpful to consider that a=b ⇔ a-b=0.

Solution
Bartleby Expert
SEE SOLUTION
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,