A functional dependency $F: X \rightarrow Y$ is termed as a $\textit{useful functional dependency}$ if and only if it satisfies all the following three conditions: - $X$ is not the empty set. - $Y$ is not the empty set. - Intersection of $X$ and $Y$ is the empty set. For a relation $R$ with 4 attributes, the total number of possible useful functional dependencies is ________

Let the set of attributes of $R$ be $U$, with $|U| = n = 4$.

A functional dependency $X \to Y$ is useful if $X \subseteq U$, $Y \subseteq U$, $X e \emptyset$, $Y e \emptyset$, and $X \cap Y = \emptyset$.

For any two disjoint subsets $X$ and $Y$ of $U$, every attribute $A \in U$ has 3 mutually exclusive possibilities:

1. $A \in X$

2. $A \in Y$

3. $A \notin X$ and $A \notin Y$

Thus, the total number of disjoint pairs $(X, Y)$ is $3^n = 3^4 = 81$.

From these, we must subtract the cases where $X = \emptyset$ or $Y = \emptyset$:

• Cases where $X = \emptyset$: $Y$ can be any subset of $U$ ($2^4 = 16$ pairs).

• Cases where $Y = \emptyset$: $X$ can be any subset of $U$ ($2^4 = 16$ pairs).

• Case where $X = \emptyset$ and $Y = \emptyset$: counted in both, so $1$ pair.

By the Principle of Inclusion-Exclusion:

Thus, the total number of possible useful functional dependencies is 50.