Let's go through each part of the problem step by step:

Part (a) Show that @ is commutative

We need to prove that $a \mathbin{@} b = b \mathbin{@} a$, where the operation $\mathbin{@}$ is defined as:

$a \mathbin{@} b = 2a + 2b$

To prove commutativity, we need to check if:

$a \mathbin{@} b = b \mathbin{@} a$

Left side: $a \mathbin{@} b$

Using the definition of the operation:

$a \mathbin{@} b = 2a + 2b$

Right side: $b \mathbin{@} a$

By the same definition, but swapping $a$ and $b$:

$b \mathbin{@} a = 2b + 2a$

Since $2a + 2b = 2b + 2a$ (addition is commutative), we have:

$a \mathbin{@} b = b \mathbin{@} a$

Thus, the operation $\mathbin{@}$ is commutative.

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Part (b) Show that @ is not associative

To prove that $\mathbin{@}$ is not associative, we need to show that:

$a \mathbin{@} (b \mathbin{@} c) \neq (a \mathbin{@} b) \mathbin{@} c$

We will first compute both sides of this equation and check if they are equal.

Left-hand side: $a \mathbin{@} (b \mathbin{@} c)$

Start by computing $b \mathbin{@} c$:

$b \mathbin{@} c = 2b + 2c$

Now, compute $a \mathbin{@} (b \mathbin{@} c)$:

$a \mathbin{@} (b \mathbin{@} c) = a \mathbin{@} (2b + 2c)$

By the definition of $\mathbin{@}$, we have:

$a \mathbin{@} (2b + 2c) = 2a + 2(2b + 2c) = 2a + 4b + 4c$

Thus, the left-hand side is:

$a \mathbin{@} (b \mathbin{@} c) = 2a + 4b + 4c$

Right-hand side: $(a \mathbin{@} b) \mathbin{@} c$

Start by computing $a \mathbin{@} b$:

$a \mathbin{@} b = 2a + 2b$

Now, compute $(a \mathbin{@} b) \mathbin{@} c$:

$(2a + 2b) \mathbin{@} c = 2(2a + 2b) + 2c = 4a + 4b + 2c$

Thus, the right-hand side is:

$(a \mathbin{@} b) \mathbin{@} c = 4a + 4b + 2c$

Comparing the two sides:

$a \mathbin{@} (b \mathbin{@} c) = 2a + 4b + 4c$ $(a \mathbin{@} b) \mathbin{@} c = 4a + 4b + 2c$

Clearly, these two expressions are not equal. Therefore, $\mathbin{@}$ is not associative.

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Summary:

• Part (a): We showed that $\mathbin{@}$ is commutative, as $a \mathbin{@} b = b \mathbin{@} a$.
• Part (b): We demonstrated that $\mathbin{@}$ is not associative, as $a \mathbin{@} (b \mathbin{@} c) \neq (a \mathbin{@} b) \mathbin{@} c$.

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Would you like further details or have any questions about the steps involved?

Here are some related questions you might find interesting:

1. Can you show that addition is commutative for real numbers?
2. Can you provide an example of another binary operation that is commutative but not associative?
3. What would happen if we modified the operation $a \mathbin{@} b$ to be $a \mathbin{@} b = a + b$?
4. How would you prove that multiplication is associative?
5. What is a real-world example of an operation that is associative but not commutative?

Tip: When working with binary operations, it's always a good idea to check both commutativity and associativity separately by choosing concrete examples to make your analysis easier.