If * is defined on Q+ such that

Answered question

2022-04-11

If * is defined on Q+ such that a * b = (ab) / 3 for a, b  Q+ , verify whether (Q+ , *) is an abelian group.

Answer & Explanation

Vasquez

Vasquez

Expert2023-04-27Added 669 answers

To verify whether (Q+, *) is an abelian group, we need to check if the following properties hold:
1. Closure
2. Associativity
3. Identity
4. Inverse
5. Commutativity
First, let's define the operation * on Q+ as:
a*b=ab3
1. Closure:
For any two elements a, b in Q+, a * b is defined as a fraction of two positive numbers, which is also a positive number. Hence, * is closed on Q+.
2. Associativity:
(a*b)*c=(ab3)*c=(ab3)c3=abc32
a*(b*c)=a*(bc3)=a(bc3)3=abc32
Therefore, * is associative on Q+.
3. Identity:
An element e in Q+ is an identity element of * if a * e = e * a = a for all a in Q+. Let's assume e is the identity element.
a*e=ae3=a
Solving for e, we get:
e=3
So, 3 is the identity element of * on Q+.
4. Inverse:
An element b in Q+ is the inverse of a in Q+ if a * b = b * a = e, where e is the identity element. Let's assume b is the inverse of a.
a*b=ab3=e=3
Solving for b, we get:
b=9a
So, the inverse of a in Q+ under * is 9a.
5. Commutativity:
To verify whether * is commutative, we need to check if a * b = b * a for all a, b in Q+.
a*b=ab3
b*a=ba3
Therefore, * is commutative on Q+.
Since * satisfies all the properties of an abelian group on Q+, we can conclude that (Q+, *) is an abelian group.

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