Question:
The number of commutative binary operations that can be defined on a set of 2 elements is:
- (a) 8
- (b) 6
- (c) 4
- (d) 2
Concept:
For a set with \( n \) elements:
- Total binary operations = \( n^{n^2} \)
- Commutative operations depend only on unordered pairs
Number of independent entries in commutative table:
\[ \frac{n(n+1)}{2} \]
So, number of commutative binary operations:
\[ n^{\frac{n(n+1)}{2}} \]
Solution:
Here \( n = 2 \)
\[ 2^{\frac{2(3)}{2}} = 2^3 = 8 \]
—Final Answer:
\[ \boxed{8} \]
Correct Option: (a)