A Transition to Advanced Mathematics
A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Chapter 6.4, Problem 8E

a.

To determine

To prove the remaining parts of theorem 6.4.1.

a.

Expert Solution
Check Mark

Explanation of Solution

Consider f:(A,)(B,×) an operation preserving map therefore,

  f(xy)=f(x)×f(y)

Consider f:AB is onto on B .

Consider is associative in A therefore, for some x,y,zA

  x(yz)=(xy)z

Consider,

  u,v,wB

As f:AB is onto on B therefore their exists x,y,zA such that

  u=f(x)v=f(y)w=f(z)

Consider,

  u×(v×w)=f(x)×(f(y)×f(z))=f(x(yz))=f((xy)z)=f(xy)×f(z)=(f(x)×f(y))×f(z)u×(v×w)=(u×v)×w

Therefore × is associative on B .

Hence the theorem is proved.

b.

To determine

To prove the remaining parts of theorem 6.4.1.

b.

Expert Solution
Check Mark

Explanation of Solution

Consider f:(A,)(B,×) an operation preserving map therefore,

  f(xy)=f(x)×f(y)

Consider f:AB is onto on B .

Let e be the identity in A therefore,

  ee=e In A

Therefore,

  f(ee)=f(e)f(e)×f(e)=f(e)f(e)×f(e)=f(e)×e',e' is the identity in B

By left cancellation law,

  f(e)=e'

Therefore, f(e) is the identity for B .

Hence the theorem is proved.

c.

To determine

To prove the remaining parts of theorem 6.4.1.

c.

Expert Solution
Check Mark

Explanation of Solution

Consider f:(A,)(B,×) an operation preserving map therefore,

  f(xy)=f(x)×f(y)

Consider f:AB is onto on B .

Consider,

  xA

Therefore,

  f(e)=f(xx1)=f(x)×f(x1)

And

  f(e)=f(x1x)=f(x1)×f(x)

As e'=f(e) is the identity in B

  f(x)×f(x1)=f(x1)×f(x)=e'

Therefore, f(x1) is the inverse of f(x) .

Thus,

  (f(x)1)=f(x1)

Hence the theorem is proved.

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Chapter 6 Solutions

A Transition to Advanced Mathematics

Ch. 6.1 - Prob. 11ECh. 6.1 - Prob. 12ECh. 6.1 - Prob. 13ECh. 6.1 - Prob. 14ECh. 6.1 - Prob. 15ECh. 6.1 - Prob. 16ECh. 6.2 - Show that each of the following algebraic...Ch. 6.2 - Prob. 2ECh. 6.2 - Prob. 3ECh. 6.2 - Prob. 4ECh. 6.2 - Prob. 5ECh. 6.2 - Prob. 6ECh. 6.2 - Prob. 7ECh. 6.2 - Prob. 8ECh. 6.2 - Prob. 9ECh. 6.2 - Prob. 10ECh. 6.2 - Prob. 11ECh. 6.2 - Prob. 12ECh. 6.2 - Prob. 13ECh. 6.2 - Prob. 14ECh. 6.2 - Prob. 15ECh. 6.2 - Prob. 16ECh. 6.2 - Prob. 17ECh. 6.2 - Prob. 18ECh. 6.3 - Prob. 1ECh. 6.3 - Prob. 2ECh. 6.3 - Prob. 3ECh. 6.3 - Prob. 4ECh. 6.3 - Prob. 5ECh. 6.3 - Prob. 6ECh. 6.3 - Prob. 7ECh. 6.3 - Prob. 8ECh. 6.3 - Prob. 9ECh. 6.3 - Prob. 10ECh. 6.3 - Prob. 11ECh. 6.3 - Prob. 12ECh. 6.3 - Prob. 13ECh. 6.3 - Prove that for every natural number m greater than...Ch. 6.3 - Prove that every subgroup of a cyclic group is...Ch. 6.3 - Prob. 16ECh. 6.3 - Prob. 17ECh. 6.4 - Prob. 1ECh. 6.4 - Prob. 2ECh. 6.4 - Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Prob. 9ECh. 6.4 - Prob. 10ECh. 6.4 - Prob. 11ECh. 6.4 - Prob. 12ECh. 6.4 - Prob. 13ECh. 6.4 - Prob. 14ECh. 6.4 - Prob. 15ECh. 6.4 - Prob. 16ECh. 6.4 - Is S3 isomorphic to 6,+? Explain.Ch. 6.4 - Prove that the relation of isomorphism is an...Ch. 6.4 - Prob. 19ECh. 6.4 - Prob. 20ECh. 6.5 - Prob. 1ECh. 6.5 - Prob. 2ECh. 6.5 - Prob. 3ECh. 6.5 - Prob. 4ECh. 6.5 - Prob. 5ECh. 6.5 - Prob. 6ECh. 6.5 - Prob. 7ECh. 6.5 - Prob. 8ECh. 6.5 - Prob. 9ECh. 6.5 - Prob. 10ECh. 6.5 - Prob. 11ECh. 6.5 - Prob. 12ECh. 6.5 - Prob. 13ECh. 6.5 - Prob. 14ECh. 6.5 - Prob. 15E
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