Question

If A = {a, b, c}, then the number of binary operations on A is

• A. 3
• B. 3⁶
• C. 3³
• D. 3⁹

Answer 1

The Correct Option is D

Given the set \(A = \{a, b, c\}\), we need to find the number of binary operations on A.

A binary operation on A is a function from \(A \times A\) to A.

The number of elements in A is |A| = 3.

The number of elements in \(A \times A\) is \(|A \times A| = |A| \times |A| = 3 \times 3 = 9\).

For each of the 9 elements in \(A \times A\), the binary operation can map it to any of the 3 elements in A.

Therefore, the total number of binary operations is \(3^{|A \times A|} = 3^9\).

Thus, the correct option is (D) \(3^9\).

Answer 2

For a set \( A = \{a, b, c\} \), the number of binary operations on \( A \) is calculated as follows: 

1. A binary operation takes two elements from the set and maps them to one element in the set. 

2. For each pair of elements from \( A \), there are 3 possible outcomes, since \( A \) has 3 elements.  Since there are 3 elements in \( A \), the total number of ordered pairs of elements is \( 3 \times 3 = 9 \). 

3. For each of these 9 pairs, there are 3 choices for the result. 

Therefore, the total number of binary operations is: \[ 3^9 \] Answer: (D) \( 3^9 \).