Represent all possible functions defined from `{alpha, beta}` to `{1,2}`.

Answer

To represent all possible functions defined from the set \(\{ \alpha, \beta \}\) to the set \(\{ 1, 2 \}\), we need to consider the definition of a function. A function from set \(A\) to set \(B\) assigns each element in set \(A\) to exactly one element in set \(B\).

### Step-by-Step Solution:

1. **Identify the Sets**:
- Let \(A = \{ \alpha, \beta \}\)
- Let \(B = \{ 1, 2 \}\)

2. **Determine the Number of Functions**:
- Each element in set \(A\) can be mapped to any element in set \(B\). Since there are 2 elements in set \(A\) and each can map to 2 elements in set \(B\), the total number of functions is given by \(2^{n}\), where \(n\) is the number of elements in set \(A\).
- Here, \(n = 2\), so the total number of functions is \(2^2 = 4\).

3. **List All Possible Functions**:
- We will represent the functions as pairs of mappings from elements in \(A\) to elements in \(B\):
- **Function 1**: \(f_1\) where \(\alpha \mapsto 1\) and \(\beta \mapsto 1\) → \(\{ (\alpha, 1), (\beta, 1) \}\)
- **Function 2**: \(f_2\) where \(\alpha \mapsto 1\) and \(\beta \mapsto 2\) → \(\{ (\alpha, 1), (\beta, 2) \}\)
- **Function 3**: \(f_3\) where \(\alpha \mapsto 2\) and \(\beta \mapsto 1\) → \(\{ (\alpha, 2), (\beta, 1) \}\)
- **Function 4**: \(f_4\) where \(\alpha \mapsto 2\) and \(\beta \mapsto 2\) → \(\{ (\alpha, 2), (\beta, 2) \}\)

4. **Summarize the Functions**:
- The four possible functions from set \(\{ \alpha, \beta \}\) to set \(\{ 1, 2 \}\) are:
1. \(f_1: \{ (\alpha, 1), (\beta, 1) \}\)
2. \(f_2: \{ (\alpha, 1), (\beta, 2) \}\)
3. \(f_3: \{ (\alpha, 2), (\beta, 1) \}\)
4. \(f_4: \{ (\alpha, 2), (\beta, 2) \}\)