Question 3 In each case below, verify whether the described algebraic structure is a group. Check the axioms and do not forget to verify whether the given object is indeed an algebraic structure! In each case, additionally check whether it is commutative – either provide a coun- terexample or argue that it it is indeed commuative (verify this axiom even if it is not a group, but only if it is an algebraic structure). Justify your answer and show all work. (i) (R²,⋆), where (x,y) ⋆ (a,b) = (x+a, y −b). (Here R² = {(z,w) : z, w ≤ R}.) (ii) (R\ {0},0), where a ob=a².b². (iii) The complex numbers of complex norm 1 under the usual complex multiplication. (iv) The set of all 2 × 2 upper-triangular matrices having determinant 1, under the usual matrix multiplication. (v) The set Q[x] of all polynomials in x with coefficients from Q and under the usual addition of polynomials.
Question 3 In each case below, verify whether the described algebraic structure is a group. Check the axioms and do not forget to verify whether the given object is indeed an algebraic structure! In each case, additionally check whether it is commutative – either provide a coun- terexample or argue that it it is indeed commuative (verify this axiom even if it is not a group, but only if it is an algebraic structure). Justify your answer and show all work. (i) (R²,⋆), where (x,y) ⋆ (a,b) = (x+a, y −b). (Here R² = {(z,w) : z, w ≤ R}.) (ii) (R\ {0},0), where a ob=a².b². (iii) The complex numbers of complex norm 1 under the usual complex multiplication. (iv) The set of all 2 × 2 upper-triangular matrices having determinant 1, under the usual matrix multiplication. (v) The set Q[x] of all polynomials in x with coefficients from Q and under the usual addition of polynomials.
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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How can I check closure, associativity, identity, inverse and commutativity for v) ?
![Question 3 In each case below, verify whether the described algebraic structure is a group.
Check the axioms and do not forget to verify whether the given object is indeed an algebraic
structure! In each case, additionally check whether it is commutative - either provide a coun-
terexample or argue that it it is indeed commuative (verify this axiom even if it is not a group,
but only if it is an algebraic structure). Justify your answer and show all work.
(i) (R²,*), where (x,y) ⋆ (a,b) = (x+a,y−b). (Here R² = {(z, w) : z, w = R}.)
(ii) (R\ {0},0), where a ob=a². b².
(iii) The complex numbers of complex norm 1 under the usual complex multiplication.
(iv) The set of all 2 × 2 upper-triangular matrices having determinant 1, under the usual matrix
multiplication.
(v) The set Q[x] of all polynomials in x with coefficients from Q and under the usual addition
of polynomials.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2521d3ac-1f2b-4213-a1c7-670c7b844752%2F63cfb376-e2a6-409d-967b-3c6d0215c4d1%2Fdsppc2b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 3 In each case below, verify whether the described algebraic structure is a group.
Check the axioms and do not forget to verify whether the given object is indeed an algebraic
structure! In each case, additionally check whether it is commutative - either provide a coun-
terexample or argue that it it is indeed commuative (verify this axiom even if it is not a group,
but only if it is an algebraic structure). Justify your answer and show all work.
(i) (R²,*), where (x,y) ⋆ (a,b) = (x+a,y−b). (Here R² = {(z, w) : z, w = R}.)
(ii) (R\ {0},0), where a ob=a². b².
(iii) The complex numbers of complex norm 1 under the usual complex multiplication.
(iv) The set of all 2 × 2 upper-triangular matrices having determinant 1, under the usual matrix
multiplication.
(v) The set Q[x] of all polynomials in x with coefficients from Q and under the usual addition
of polynomials.
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