To prove that Q* is a group under *, we need to show that it satisfies the four group axioms: closure, associativity, identity, and invertibility.

1. Closure: For any two elements a, b in Q*, a*b = ab/3 is also in Q* because the product and division of non-zero rational numbers is also a non-zero rational number.

2. Associativity: For any three elements a, b, c in Q*, we need to show that (a*b)*c = a*(b*c).
(a*b)*c = (ab/3)*c = abc/3^2
a*(b*c) = a*(bc/3) = abc/3^2
Therefore, (a*b)*c = a*(b*c), so * is associative.

3. Identity: We need to find an element e in Q* such that for every element a in Q*, a*e = e*a = a. Let's try e = 3.
a*3 = a3/3 = a and 3*a = 3a/3 = a. So, e = 3 is the identity element.

4. Invertibility: For each element a in Q*, there must exist an element b in Q* such that a*b = b*a = e.
Let's try b = 1/a. Then a*b = a*(1/a) = a/a = 1 and b*a = (1/a)*a = a/a = 1. So, b = 1/a is the inverse of a.

Therefore, Q* is a group under *.

Question

To prove that Q* is a group under *, we need to show that it satisfies the four group axioms: closure, associativity, identity, and invertibility.

1. Closure: For any two elements a, b in Q*, a*b = ab/3 is also in Q* because the product and division of non-zero rational numbers is also a non-zero rational number.

2. Associativity: For any three elements a, b, c in Q*, we need to show that (a*b)*c = a*(b*c). 
   (a*b)*c = (ab/3)*c = abc/3^2
   a*(b*c) = a*(bc/3) = abc/3^2
   Therefore, (a*b)*c = a*(b*c), so * is associative.

3. Identity: We need to find an element e in Q* such that for every element a in Q*, a*e = e*a = a. Let's try e = 3.
   a*3 = a3/3 = a and 3*a = 3a/3 = a. So, e = 3 is the identity element.

4. Invertibility: For each element a in Q*, there must exist an element b in Q* such that a*b = b*a = e. 
   Let's try b = 1/a. Then a*b = a*(1/a) = a/a = 1 and b*a = (1/a)*a = a/a = 1. So, b = 1/a is the inverse of a.

Therefore, Q* is a group under *.

Knowee AI · Accepted Answer

To prove that Q* is a group under *, we need to show that it satisfies the four group axioms: closure, associativity, identity, and invertibility.

1. Closure: For any two elements a, b in Q*, a*b = ab/3 is also in Q* because the product and division of non-zero rational numbers is also a non-zero rational number.

2. Associativity: For any three elements a, b, c in Q*, we need to show that (a*b)*c = a*(b*c). 
   (a*b)*c = (ab/3)*c = abc/3^2
   a*(b*c) = a*(bc/3) = abc/3^2
   Therefore, (a*b)*c = a*(b*c), so * is associative.

3. Identity: We need to find an element e in Q* such that for every element a in Q*, a*e = e*a = a. Let's try e = 3.
   a*3 = a3/3 = a and 3*a = 3a/3 = a. So, e = 3 is the identity element.

4. Invertibility: For each element a in Q*, there must exist an element b in Q* such that a*b = b*a = e. 
   Let's try b = 1/a. Then a*b = a*(1/a) = a/a = 1 and b*a = (1/a)*a = a/a = 1. So, b = 1/a is the inverse of a.

Therefore, Q* is a group under *.

Let 𝑄∗ be the set of non-zero rational numbers and * be the binary operation on Q definedby 𝑎 ∗ 𝑏 = 𝑎𝑏/3. Prove that 𝑄∗ is group under *.

Question

Let $Q^$ be the set of non-zero rational numbers and $$ be the binary operation on $Q$ defined by $a * b = \frac{ab}{3}$ . Prove that $Q^$ is a group under $$ .

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