Construction a composition table for binary operation `^^` defined as ` a ^^ b`= minimum of {a,b} in the set {1,2,3,4,5} and
(i) evaluate `(2 ^^ 3) ^^4 and 2^^ (3^^ 4)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Construct the Composition Table

The binary operation \( a \, ^^ \, b \) is defined as the minimum of \( a \) and \( b \) for \( a, b \in \{1, 2, 3, 4, 5\} \).

We will create a table where the rows and columns represent the elements of the set, and each cell will contain the result of the operation \( a \, ^^ \, b \).

| ^^ | 1 | 2 | 3 | 4 | 5 |
|-----|---|---|---|---|---|
| **1** | 1 | 1 | 1 | 1 | 1 |
| **2** | 1 | 2 | 2 | 2 | 2 |
| **3** | 1 | 2 | 3 | 3 | 3 |
| **4** | 1 | 2 | 3 | 4 | 4 |
| **5** | 1 | 2 | 3 | 4 | 5 |

### Step 2: Evaluate \( (2 \, ^^ \, 3) \, ^^ \, 4 \)

1. First, we need to calculate \( 2 \, ^^ \, 3 \):
\[ 2 \, ^^ \, 3 = \min(2, 3) = 2 \]

2. Next, we use the result from the first step to calculate \( 2 \, ^^ \, 4 \):
\[ (2 \, ^^ \, 3) \, ^^ \, 4 = 2 \, ^^ \, 4 = \min(2, 4) = 2 \]

### Step 3: Evaluate \( 2 \, ^^ \, (3 \, ^^ \, 4) \)

1. First, we need to calculate \( 3 \, ^^ \, 4 \):
\[ 3 \, ^^ \, 4 = \min(3, 4) = 3 \]

2. Next, we use the result from the first step to calculate \( 2 \, ^^ \, 3 \):
\[ 2 \, ^^ \, (3 \, ^^ \, 4) = 2 \, ^^ \, 3 = \min(2, 3) = 2 \]

### Final Results

Both evaluations yield the same result:
\[ (2 \, ^^ \, 3) \, ^^ \, 4 = 2 \quad \text{and} \quad 2 \, ^^ \, (3 \, ^^ \, 4) = 2 \]

Thus, the final answer is:
\[ \text{Both expressions evaluate to } 2. \]

---