6. a) A binary operation * is defined on the set of real numbers as follows: $$\na * b=2^{-a}+b, \quad a, b \in \mathbf{R} $$ (i) Is the operation * commutative? If not give a counter example. (ii) Find the value of $$ -1 *(0 * 1) $$ and $$ (-1 * 0) * 1 $$, and state whether $$ * $$ is associative.

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Let's analyze the binary operation $$ * $$ defined on the set of real numbers by:
$$
a * b = 2^{-a} + b, \quad \text{for all } a, b \in \mathbb{R}
$$

### (i) Commutativity of $$ * $$

A binary operation $$ * $$ is **commutative** if for all $$ a, b \in \mathbb{R} $$,
$$
a * b = b * a
$$
Let's check whether this holds for our operation.

**Example:**
Let $$ a = 0 $$ and $$ b = 1 $$.

- Compute $$ 0 * 1 $$:
  $$
  0 * 1 = 2^{-0} + 1 = 1 + 1 = 2
  $$
  
- Compute $$ 1 * 0 $$:
  $$
  1 * 0 = 2^{-1} + 0 = \frac{1}{2} + 0 = \frac{1}{2}
  $$
  
Since $$ 0 * 1 = 2 $$ and $$ 1 * 0 = \frac{1}{2} $$, we have:
$$
0 * 1 \neq 1 * 0
$$
**Conclusion:** The operation $$ * $$ is **not commutative**. The values $$ 0 * 1 $$ and $$ 1 * 0 $$ serve as a counterexample.

---

### (ii) Associativity of $$ * $$

A binary operation $$ * $$ is **associative** if for all $$ a, b, c \in \mathbb{R} $$,
$$
a * (b * c) = (a * b) * c
$$
Let's evaluate both $$ -1 * (0 * 1) $$ and $$ (-1 * 0) * 1 $$ to check associativity.

**Step 1:** Compute $$ 0 * 1 $$:
$$
0 * 1 = 2^{-0} + 1 = 1 + 1 = 2
$$

**Step 2:** Compute $$ -1 * (0 * 1) = -1 * 2 $$:
$$
-1 * 2 = 2^{-(-1)} + 2 = 2^{1} + 2 = 2 + 2 = 4
$$

**Step 3:** Compute $$ -1 * 0 $$:
$$
-1 * 0 = 2^{-(-1)} + 0 = 2^{1} + 0 = 2 + 0 = 2
$$

**Step 4:** Compute $$ (-1 * 0) * 1 = 2 * 1 $$:
$$
2 * 1 = 2^{-2} + 1 = \frac{1}{4} + 1 = \frac{5}{4}
$$

**Comparison:**
$$
-1 * (0 * 1) = 4 \quad \text{and} \quad (-1 * 0) * 1 = \frac{5}{4}
$$
Since $$ 4 \neq \frac{5}{4} $$, the operation $$ * $$ is **not associative**.

---

### **Final Summary:**
1. **Commutativity:** The operation $$ * $$ is **not commutative**. For example, $$ 0 * 1 = 2 $$ while $$ 1 * 0 = \frac{1}{2} $$.
2. **Associativity:** The operation $$ * $$ is **not associative**. Specifically, $$ -1 * (0 * 1) = 4 $$ differs from $$ (-1 * 0) * 1 = \frac{5}{4} $$.
The operation * is **not commutative** because $$ 0 * 1 \neq 1 * 0 $$. Additionally, * is **not associative** since $$ -1 * (0 * 1) \neq (-1 * 0) * 1 $$.

6. a) A binary operation * is defined on the set of real numbers as follows: \[\na * b=2^{-a}+b, \quad a, b \in \mathbf{R} \] (i) Is the operation * commutative? If not give a counter example. (ii) Find the value of \( -1 (0 1) \) and \( (-1 * 0) * 1 \), and state whether \( * \) is associative.

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6. a) A binary operation * is defined on the set of real numbers as follows: \[\na * b=2^{-a}+b, \quad a, b \in \mathbf{R} \] (i) Is the operation * commutative? If not give a counter example. (ii) Find the value of \( -1 *(0 * 1) \) and \( (-1 * 0) * 1 \), and state whether \( * \) is associative.