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For each binary operation * defined on a set below, determine whether or not * gives a group structure on the set. If it is a group, explain why it is a group. If it is not a group, say which axioms fail to hold: (a) Define * on Z by a * b max(a, b). %3D (b) Define * on Q \ {0} by a * b = |ab|. (c) Define * on Q+ by a * b = ab.

For each binary operation * defined on a set below, determine whether or not * gives a group structure on the set. If it is a group, explain why it is a group. If it is not a group, say which axioms fail to hold: (a) Define * on Z by a * b max(a, b). %3D (b) Define * on Q \ {0} by a * b = |ab|. (c) Define * on Q+ by a * b = ab.

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
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For each binary operation \( \ast \) defined on a set below, determine whether or not \( \ast \) gives a group structure on the set. If it is a group, explain why it is a group. If it is not a group, say which axioms fail to hold: (a) Define \( \ast \) on \( \mathbb{Z} \) by \( a \ast b = \max(a, b) \). (b) Define \( \ast \) on \( \mathbb{Q} \setminus \{0\} \) by \( a \ast b = |ab| \). (c) Define \( \ast \) on \( \mathbb{Q}^+ \) by \( a \ast b = ab \).
Transcribed Image Text:For each binary operation \( \ast \) defined on a set below, determine whether or not \( \ast \) gives a group structure on the set. If it is a group, explain why it is a group. If it is not a group, say which axioms fail to hold: (a) Define \( \ast \) on \( \mathbb{Z} \) by \( a \ast b = \max(a, b) \). (b) Define \( \ast \) on \( \mathbb{Q} \setminus \{0\} \) by \( a \ast b = |ab| \). (c) Define \( \ast \) on \( \mathbb{Q}^+ \) by \( a \ast b = ab \).
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