Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(c) Let G3 be the set of all nonconstant functions from R to R of the form f(r) = ax +b where a is a nonzero real number and b can be any real number. Prove or disprove: G3 is a group under composition. (d) Let G4 be the set of all functions from R to R of the form f(x) ar³,where a is a nonzero real number. Prove or disprove: G4 is a group under composition. %3D

(c) Let G3 be the set of all nonconstant functions from R to R of the form f(r) = ax +b where a is a nonzero real number and b can be any real number. Prove or disprove: G3 is a group under composition. (d) Let G4 be the set of all functions from R to R of the form f(x) ar³,where a is a nonzero real number. Prove or disprove: G4 is a group under composition. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please do exercise 8.8.4 and do part c and d and please show step by step and explain

Proposition 8.8.3. Let A be a set, and let G be the set of all invertible functions from A to A. Then G is a group under composition. In the following exercise we look at some particular sets of functions, and investigate whether or not these sets form groups under composition. Recall that to show whether or not a set with binary operation is a group, you just need to show the properties: closure, identity, inverse, and associative. We're lucky in this case that we don't have to prove associative in every single case, because the operation of function composition is always associative, as we've proven before. So it's enough just to prove closure, identity, and inverse. Exercise 8.8.4. (a) Let G1 be the set of all nonzero functions from R to R of the form = ax, where a is a nonzero real number. (For example, the func- tions g(r) = -7x and h(x) = v2x are both elements of G1.) Prove or disprove: G1 is a group under composition. (Note: G1 is the set of nonzero linear functions from R to R.) (b) Let G2 be the set of all nonzero functions from R to R of the form f(x) zero. (For example, the functions p(x) = 29.4x + 42.3, q(x) = 15 and r(x) = -nx are all elements of G2.) Prove or disprove: G2 is a group under composition. (Note: G2 is called the set of all nonzero affine functions from R to R.) = ax + b where a and b are real numbers which are not both (c) Let G3 be the set of all nonconstant functions from R to R of the form f(r) = ax + b where a is a nonzero real number and b can be any real number. Prove or disprove: G3 is a group under composition. (d) Let G4 be the set of all functions from R to R of the form f(x) = ax3,where a is a nonzero real number. Prove or disprove: G4 is a group under composition.
Transcribed Image Text:Proposition 8.8.3. Let A be a set, and let G be the set of all invertible functions from A to A. Then G is a group under composition. In the following exercise we look at some particular sets of functions, and investigate whether or not these sets form groups under composition. Recall that to show whether or not a set with binary operation is a group, you just need to show the properties: closure, identity, inverse, and associative. We're lucky in this case that we don't have to prove associative in every single case, because the operation of function composition is always associative, as we've proven before. So it's enough just to prove closure, identity, and inverse. Exercise 8.8.4. (a) Let G1 be the set of all nonzero functions from R to R of the form = ax, where a is a nonzero real number. (For example, the func- tions g(r) = -7x and h(x) = v2x are both elements of G1.) Prove or disprove: G1 is a group under composition. (Note: G1 is the set of nonzero linear functions from R to R.) (b) Let G2 be the set of all nonzero functions from R to R of the form f(x) zero. (For example, the functions p(x) = 29.4x + 42.3, q(x) = 15 and r(x) = -nx are all elements of G2.) Prove or disprove: G2 is a group under composition. (Note: G2 is called the set of all nonzero affine functions from R to R.) = ax + b where a and b are real numbers which are not both (c) Let G3 be the set of all nonconstant functions from R to R of the form f(r) = ax + b where a is a nonzero real number and b can be any real number. Prove or disprove: G3 is a group under composition. (d) Let G4 be the set of all functions from R to R of the form f(x) = ax3,where a is a nonzero real number. Prove or disprove: G4 is a group under composition.
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
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,
SEE MORE TEXTBOOKS