Let's follow your requested steps: --- ### **Step 1: Introduction of the Concept** A **functional dependency** (FD) in a relation means that for any two rows, if the values for the left-hand side attributes are the same, then the values for the right-hand side attributes must also be the same. For example, \( X \rightarrow Y \) means that if two rows have the same \( X \), they must also have the same \( Y \). --- ### **Step 2: Correct Answer and Explanation** #### **Finding the correct FD(s):** Let's analyze each FD: Given Relation S: | X | Y | Z | |---|----|---| | m | 20 | T | | m | 10 | F | | o | 30 | T | | n | 30 | T | | o | 20 | T | #### a. **X → Y** - Check rows where X is the same: - For X = m: (20, T) and (10, F) → Y changes (20 ≠ 10) - For X = o: (30, T) and (20, T) → Y changes (30 ≠ 20) - **Does not hold.** #### b. **Z → Y** - For Z = T: (20, T), (30, T), (30, T), (20, T) → Y = 20 or 30 → Y changes - **Does not hold.** #### c. **Y → Z** - For Y = 20: (m, 20, T), (o, 20, T) → Z = T for both. - For Y = 10: (m, 10, F) → Z = F. - For Y = 30: (o, 30, T), (n, 30, T) → Z = T for both. - **Holds.** #### d. **XY → Z** - Check pairs: - (m, 20): (m, 20, T) only one row. - (m, 10): (m, 10, F) only one row. - (o, 30): (o, 30, T) only one row. - (n, 30): (n, 30, T) only one row. - (o, 20): (o, 20, T) only one row. - All combinations are unique; thus, it holds **trivially**. #### e. **YZ → X** - (20, T): (m, 20, T), (o, 20, T) → X = m, o → X changes. - (10, F): (m, 10, F) → X = m - (30, T): (o, 30, T), (n, 30, T) → X = o, n → X changes. - **Does not hold.** #### f. **XZ → Y** - (m, T): (m, 20, T) - (m, F): (m, 10, F) - (o, T): (o, 30, T), (o, 20, T) → Y = 30, 20 → Y changes. - (n, T): (n, 30, T) - **Does not hold.** **Correct answers: (c) Y → Z and (d) XY → Z** --- **Explanation:** - **Y → Z holds** because for every same value of Y, Z is always the same. - **XY → Z holds trivially** because every combination of (X, Y) is unique, so for the same (X, Y) pair, Z will always be the same. --- ### **Step 3: Why Other Options Are Not Correct** - **a. X → Y:** For the same X, Y can be different (see X = m and X = o). - **b. Z → Y:** For the same Z, Y can be different (see Z = T). - **e. YZ → X:** For the same (Y, Z), X can be different (see (30, T) and (20, T)). - **f. XZ → Y:** For the same (X, Z), Y can be different (see (o, T)). --- **Final Answer: (c) Y → Z and (d) XY → Z** --- ### **Summary of Steps (one sentence each):** - **Step 1:** Introduced the concept of functional dependency. - **Step 2:** Identified which dependencies hold and explained why. - **Step 3:** Explained why the other options do not hold and provided the final answer.