Question

A functional dependency \(F: X \to Y\) is termed as a 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 ............

Hint:

Useful functional dependencies ensure \(X\) and \(Y\) are disjoint, non-empty subsets.

Answer 1

For a relation \(R\) with 4 attributes \(\{A, B, C, D\}\), we need to compute the total number of useful functional dependencies \(X \to Y\), where: \(X\) is a non-empty subset of the attributes. \(Y\) is a non-empty subset of the attributes. \(X\) and \(Y\) are disjoint (\(X \cap Y = \emptyset\)). Step 1: Total number of subsets of \(R\) The total number of subsets of \(\{A, B, C, D\}\) is \(2^4 = 16\). These subsets include: \[ \{\emptyset, \{A\}, \{B\}, \{C\}, \{D\}, \{A, B\}, \{A, C\}, \ldots, \{A, B, C, D\}\}. \] Excluding the empty set, there are \(15\) non-empty subsets.
Step 2: Counting valid combinations of \(X\) and \(Y\) Since \(X \cap Y = \emptyset\), \(Y\) must be chosen from the attributes that are not in \(X\). For a fixed \(X\): The size of \(X\) determines the attributes available for \(Y\). If \(X\) contains \(k\) attributes, then \(4 - k\) attributes are available for \(Y\). For every non-empty subset \(X\) (15 options), the number of possible non-empty subsets of the remaining attributes (\(Y\)) is: \[ 2^{4 - |X|} - 1. \]
Step 3: Summing over all valid combinations To find the total number of useful functional dependencies, sum over all non-empty subsets \(X\): \[ \text{Total functional dependencies} = \sum_{k=1}^{4} \binom{4}{k} \cdot (2^{4 - k} - 1), \] where \(\binom{4}{k}\) is the number of ways to choose \(k\) attributes for \(X\).
Step 4: Calculate each term For \(k = 1, 2, 3, 4\): For \(k = 1\): \[ \binom{4}{1} \cdot (2^{4 - 1} - 1) = 4 \cdot (8 - 1) = 4 \cdot 7 = 28. \] For \(k = 2\): \[ \binom{4}{2} \cdot (2^{4 - 2} - 1) = 6 \cdot (4 - 1) = 6 \cdot 3 = 18. \] For \(k = 3\): \[ \binom{4}{3} \cdot (2^{4 - 3} - 1) = 4 \cdot (2 - 1) = 4 \cdot 1 = 4. \] For \(k = 4\): \[ \binom{4}{4} \cdot (2^{4 - 4} - 1) = 1 \cdot (1 - 1) = 0. \] Step 5: Add the results The total number of useful functional dependencies is: \[ 28 + 18 + 4 + 0 = 50. \] Final Answer: \[ \boxed{50} \]