The number of binary operations that can be defined on a set of `2` elements is

Options

A. (a) `8`
B. (b) `4`
C. (c) `16`
D. (d) `64`

Explanation

To find the number of binary operations that can be defined on a set of 2 elements, we can follow these steps:

### Step 1: Understand the Definition of Binary Operation
A binary operation on a set is a function that combines two elements from the set to produce another element from the same set.

### Step 2: Identify the Set Size
Given that the set has 2 elements, we can denote the set as \( S = \{a, b\} \).

### Step 3: Determine the Number of Possible Outputs
For a binary operation, we need to consider all possible pairs of elements from the set. Since there are 2 elements, the pairs can be:
- \( (a, a) \)
- \( (a, b) \)
- \( (b, a) \)
- \( (b, b) \)

This gives us a total of \( 2 \times 2 = 4 \) pairs.

### Step 4: Calculate the Number of Binary Operations
For each of these 4 pairs, we can choose any of the 2 elements from the set as the output. Therefore, for each pair, we have 2 choices.

The total number of binary operations can be calculated as:
\[ \text{Number of binary operations} = \text{(number of pairs)}^{\text{(number of choices per pair)}} \]
This can be expressed mathematically as:
\[ \text{Number of binary operations} = n^{n^2} \]
where \( n \) is the number of elements in the set.

### Step 5: Substitute the Values
Here, \( n = 2 \):
\[ \text{Number of binary operations} = 2^{2^2} = 2^4 = 16 \]

### Conclusion
Thus, the number of binary operations that can be defined on a set of 2 elements is **16**.

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