5. (a)Decide whether (Z, -) forms a group where : Z x Z Zis the usual operation of subtraction, i.e. (m, n) m - n. Justify youranswer fully.X1(b)Consider R? = R x R. Elements of R? have the forminwhich x1, x2 E R. Define the operation : R2 x R2 R? by) - „).y2\Y1x2+ y2,Show that (R?, *) does not form a group. Next, find a suitable subset ofR? that forms a group with the operation * as given above. Finally, decidewith proof whether this group is abelian.Show (R\{-1}, o) forms a group, where for any a, be R\{-1},(c)a ob = (a+1)(b+1) – 1. Next, find a group element g such that 2og o3 =5. Is this element unique? Justify your answer.

NOTE: According to guideline answer of first question can be given, for other please ask in a…

Answered: Decide whether (Z, -) forms a group…

Decide whether (Z, -) forms a group where : Z xZ Z (a) is the usual operation of subtraction, i.e. (m, n) m - n. Justify your answer fully.

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

5. (a)
Decide whether (Z, -) forms a group where : Z x Z Z
is the usual operation of subtraction, i.e. (m, n) m - n. Justify your
answer fully.
X1
(b)
Consider R? = R x R. Elements of R? have the form
in
which x1, x2 E R. Define the operation : R2 x R2 R? by
) - „).
y2
\Y1x2+ y2,
Show that (R?, *) does not form a group. Next, find a suitable subset of
R? that forms a group with the operation * as given above. Finally, decide
with proof whether this group is abelian.
Show (R\{-1}, o) forms a group, where for any a, be R\{-1},
(c)
a ob = (a+1)(b+1) – 1. Next, find a group element g such that 2og o3 =
5. Is this element unique? Justify your answer.

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