The image presents a binary operation defined on the set of integers $\mathbb{Z}$, given by:

$a \circ b = -2ab$

You are being asked the following three questions about this operation:

1. Is the operation commutative?
2. Is the operation associative?
3. Does the operation have a neutral (identity) element?

Let's analyze each of these properties.

1. Commutativity

An operation is commutative if $a \circ b = b \circ a$ for all $a$ and $b$ in $\mathbb{Z}$.

For the operation $a \circ b = -2ab$, let's check commutativity:

$a \circ b = -2ab$ $b \circ a = -2ba$

Since multiplication of integers is commutative ($ab = ba$), we have $-2ab = -2ba$. Therefore, the operation is commutative.

2. Associativity

An operation is associative if $(a \circ b) \circ c = a \circ (b \circ c)$ for all $a$, $b$, and $c$ in $\mathbb{Z}$.

Let's check this for $a \circ b = -2ab$.

First, calculate $(a \circ b) \circ c$: $a \circ b = -2ab, \quad (a \circ b) \circ c = (-2ab) \circ c = -2(-2ab)c = 4abc$

Now, calculate $a \circ (b \circ c)$: $b \circ c = -2bc, \quad a \circ (b \circ c) = a \circ (-2bc) = -2a(-2bc) = 4abc$

Since $(a \circ b) \circ c = a \circ (b \circ c) = 4abc$, the operation is associative.

3. Neutral (Identity) Element

An operation has a neutral element $e$ if $a \circ e = a$ for all $a \in \mathbb{Z}$.

For the operation $a \circ e = -2ae = a$, solving for $e$: $-2ae = a \implies e = -\frac{1}{2}$

However, $e = -\frac{1}{2}$ is not an integer, and since the operation is defined on $\mathbb{Z}$, the set of integers, there is no neutral element in $\mathbb{Z}$. Therefore, the operation does not have a neutral element.

Summary:

1. The operation is commutative.
2. The operation is associative.
3. The operation does not have a neutral element.

Would you like further details or clarifications? Here are five follow-up questions related to this:

1. What happens if we define this operation on the set of rational numbers $\mathbb{Q}$?
2. How does this operation compare to the standard multiplication operation in $\mathbb{Z}$?
3. What are some examples of non-associative operations?
4. Could an identity element exist if we extend the set to non-integers?
5. How would we verify if an operation is distributive over addition?

Tip: When verifying commutativity or associativity, always expand both sides of the equation independently before comparing them.