please just answer d. 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 $.(d) Define $ \ast $ on $ \mathbb{Q}^+ $ by $ a \ast b = a/b $.

Answered: For each binary operation * defined on…

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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please just answer d.

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 $.

(d) Define $ \ast $ on $ \mathbb{Q}^+ $ by $ a \ast b = a/b $.

Expert Solution

Step 1

Properties of a group:

Consider the dot (*) to be an operation and the letter G to be a group. The axioms of group theory are defined as follows:

Conclusion: If x and y are two distinct elements in group G, then x.y is likewise a member of the group G.
Associativity: If x,y and z are elements in group G, then $x * (y * z) = (x * y) * z$ .

Invertibility: There exists some y in the group G for every element x in the group G in such a way that; $x y = y x$ .
Identity: For each given element x in the group G, there exists another element in the group G named I in such a way that $x * I = I * x$ , where I refers to the group G's identity element.