Let's break down the problem step by step.

Part (a): Show that # is associative.

Definition of #: According to the problem, the operation $a \# b$ simply equals $a$. This is a constant function that always returns the first operand, no matter what the second operand is.

To prove that # is associative, we need to show that:

$(a \# b) \# c = a \# (b \# c)$

Step 1: Calculate $(a \# b) \# c$

Using the definition of #: $a \# b = a$ Thus: $(a \# b) \# c = a \# c = a$

Step 2: Calculate $a \# (b \# c)$

Using the definition of # again: $b \# c = b$ So: $a \# (b \# c) = a \# b = a$

Conclusion for (a): Since both sides are equal, we have: $(a \# b) \# c = a \# (b \# c)$ Thus, # is associative.

---

Part (b): Show that # is not commutative.

Definition of commutativity: An operation is commutative if: $a \# b = b \# a$ for all values of $a$ and $b$.

Let's test this condition.

Step 1: Calculate $a \# b$

Using the definition: $a \# b = a$

Step 2: Calculate $b \# a$

Using the definition: $b \# a = b$

Step 3: Compare $a \# b$ and $b \# a$

For the operation to be commutative, we need: $a \# b = b \# a$ But we know: $a \# b = a \quad \text{and} \quad b \# a = b$ These are only equal if $a = b$. Therefore, the operation $\#$ is not commutative because in general, $a \neq b$.

---

Final Answer:

• (a) # is associative because $(a \# b) \# c = a \# (b \# c)$ for all $a$, $b$, and $c$.
• (b) # is not commutative because $a \# b \neq b \# a$ unless $a = b$.

---

Do you need more details or have any questions? Here are some related questions to deepen the understanding:

1. Can an operation that always returns the first operand be both commutative and associative?
2. How do we define associativity and commutativity more formally in abstract algebra?
3. What are some examples of non-commutative operations in mathematics?
4. How would the operation $a \# b = \min(a, b)$ behave with respect to associativity and commutativity?
5. Can an operation that is not commutative still be useful in mathematical structures, such as group theory?

Tip: Associativity and commutativity are fundamental properties in many algebraic structures like groups, rings, and fields. These properties help simplify calculations and prove important theorems!