HW2 Problem 6

Problem 6

Let $S$ be a set having exactly one element. How many different binary operations can be defined on $S$? Answer the question if $S$ has exactly 2 elements; exactly 3 elements; exactly n elements.

Solution

If $S$ has only one element then it also has only one binary operation because the only outcome possible is that element.
If $S$ has exactly 2 elements: $2$ options for $4$ places = $2^4=16$
If $S$ has exactly 3 elements: $3$ options for $9$ places = $3^9=19,683$
If $S$ has exactly n elements: $n$ options for $n^2$ places = $n^{n^2}$