Final answer:
The neutral element for the operation defined on the set of real numbers is 3/2. After investigating the operation for finding an inverse element, it becomes apparent that there is no real number that can serve as an inverse for every element of the set. Thus, there are no inverse elements.
Step-by-step explanation:
To find the neutral element and inverse element for the operation * defined by a*b = 2a/2 + 2b/2 - 3/2 for all a, b ∈ R, we first define what these terms mean in the context of this operation. A neutral element e is one for which a*e = e*a = a for all a. An inverse element for a specific a is an element a' such that a*a' = a'*a = e.
Let's find the neutral element e by setting a*e = a:
- a*e = 2a/2 + 2e/2 - 3/2
- To be a neutral element, this should equal a, so a = 2a/2 + 2e/2 - 3/2
- After simplification, we get e = 3/2 because the terms involving a will cancel out.
Now to find the inverse element a' for a, we set a*a' = e:
- a*a' = 2a/2 + 2a'/2 - 3/2
- Setting this equal to the neutral element e (3/2), we get 3/2 = 2a/2 + 2a'/2 - 3/2
- Upon solving for a', we find that there is no real number that satisfies this equation for every value of a. Hence, no inverse elements exist.
Therefore, the correct answer is (C) The neutral element is 3/2, and there are no inverse elements.