Number of binary operations on the set {a, b} are
(A) 10              (B) 16
(C) 20            (D) 8

Explanation

To find the number of binary operations on the set {a, b}, we can follow these steps:

### Step 1: Identify the set and its elements
The given set is {a, b}. This set has 2 elements.

### Step 2: Determine the number of elements in the set
Let \( n \) be the number of elements in the set. Here, \( n = 2 \).

### Step 3: Calculate the number of pairs in the Cartesian product
A binary operation on a set involves taking two elements from the set and combining them. The number of pairs that can be formed from the set is given by the Cartesian product \( S \times S \). Since there are \( n \) elements in the set, the number of pairs is:
\[ n^2 = 2^2 = 4 \]
This means there are 4 pairs: (a, a), (a, b), (b, a), (b, b).

### Step 4: Determine the number of choices for the operation
For each of these pairs, we can choose any element from the set {a, b} as the result of the operation. Since there are 2 choices for each pair, we can calculate the total number of binary operations.

### Step 5: Calculate the total number of binary operations
The total number of binary operations is given by:
\[ \text{Number of binary operations} = (\text{number of elements in the codomain})^{(\text{number of elements in the domain})} \]
In this case, the number of elements in the codomain is \( n = 2 \) and the number of elements in the domain is \( n^2 = 4 \). Therefore, we have:
\[ \text{Number of binary operations} = 2^{4} = 16 \]

### Conclusion
Thus, the number of binary operations on the set {a, b} is 16, which corresponds to option (B).

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