Find the number of binary operations that can be defined on the set A={a,b,c}

Answer

To find the number of binary operations that can be defined on the set \( A = \{a, b, c\} \), we can follow these steps:

### Step 1: Understand the Definition of a Binary Operation
A binary operation on a set \( A \) is a function that combines two elements from the set to produce another element from the same set. We denote this operation as \( * \).

### Step 2: Determine the Domain and Codomain
The binary operation \( * \) can be defined as a function from \( A \times A \) (the Cartesian product of \( A \) with itself) to \( A \). This means we need to consider pairs of elements from \( A \).

### Step 3: Calculate the Size of \( A \)
The set \( A \) has 3 elements: \( a, b, c \). Therefore, the cardinality of \( A \) is \( |A| = 3 \).

### Step 4: Calculate the Size of \( A \times A \)
The Cartesian product \( A \times A \) consists of all possible ordered pairs of elements from \( A \). The number of elements in \( A \times A \) is given by:
\[ |A \times A| = |A| \times |A| = 3 \times 3 = 9 \]

### Step 5: Determine the Number of Functions from \( A \times A \) to \( A \)
For each of the 9 pairs in \( A \times A \), we can choose any of the 3 elements in \( A \) as the output of the function. Therefore, the total number of functions (binary operations) from \( A \times A \) to \( A \) is given by:
\[ \text{Number of binary operations} = |A|^{|A \times A|} = 3^{9} \]

### Step 6: Calculate \( 3^9 \)
Now, we compute \( 3^9 \):
\[ 3^9 = 19683 \]

### Final Answer
Thus, the total number of binary operations that can be defined on the set \( A = \{a, b, c\} \) is \( 19683 \).

---