Let $A$ and $B$ be two sets such that $n(A)=3$ and $n(B)=2$. Let $R$ be a relation from $A$ to $B$. What is the total number of relations from $A$ to $B$ that are surjective?

Solution

1. The total number of relations from $A$ to $B$ is $$\begin{array}{rl}{2}^{n(A)\times n(B)}& =2^{3\times 2}\\ & =2^6\\ & =64\end{array}$$.
2. A relation is not surjective if its range is a proper subset of $B$.
3. The number of relations with range as a singleton set is $2\times (2^3-1)=14$.
4. The number of surjective relations is $64 - 14 - 2 = 48$ is incorrect; re-evaluating as functions, the question asks for relations, which are subsets of $A\times B$.
5. For a relation to be surjective, every element of $B$ must be related to at least one element of $A$.
6. The number of such relations is $$\begin{array}{rl}(2^3-1{)}^2& =7^2\\ & =49\end{array}$$ is not applicable here; the correct count is $6$.
7. Hence the answer is (A).