A functional dependency is a constraint between two sets of attributes within a relation. If A and B are sets of attributes in a relation R, we say that A functionally determines B (written as A→B) if, for any two tuples (rows) in R, whenever their values for A are identical, their values for B must also be identical.

In simpler terms: If you know the value(s) of attribute(s) in set A, you can uniquely determine the value(s) of attribute(s) in set B. The attributes in A are called the determinant, and the attributes in B are called the dependent.

Let's look at our Student_Course_Instructor table and identify some FDs:

Here are some functional dependencies that naturally exist based on the meaning of the data:

• StudentID → StudentName, StudentMajor
  Explanation: If you know a StudentID, you uniquely know that student's Name and Major. (Alice (S101) is always 'Alice' and her major is 'CS').
• CourseID → CourseTitle
  Explanation: If you know a CourseID, you uniquely know its Title. (CS101 is always 'Intro to DB').
• InstructorName → InstructorDept
  Explanation: If you know an InstructorName, you uniquely know their Department. (Prof. Smith is always in 'Computer Sci.').
• (CourseID, InstructorName) → InstructorDept
  Explanation: If you know the CourseID and InstructorName, you know the InstructorDept.
• (StudentID, CourseID) → InstructorName, InstructorDept
  Explanation: The combination of a StudentID and CourseID determines which Instructor teaches that student in that course, and by extension, that instructor's Department. This implies that (StudentID, CourseID) is a candidate key for this particular table.

Functional dependencies are categorized based on the relationship between their dependent and determinant sets:

• Trivial Functional Dependency: A functional dependency A→B is trivial if B is a subset of A (i.e., B⊆A). In simpler terms, if the attributes on the right-hand side (the dependent) are already included in the attributes on the left-hand side (the determinant), then the dependency is trivially true. You inherently know B if you know A because B is part of A.
• Example:
  • StudentID, CourseID → StudentID (because StudentID is part of (StudentID, CourseID))
  • InstructorName, InstructorDept → InstructorName
  • StudentID → StudentID (This is the simplest trivial FD)
  • Trivial FDs are always true for any relation and don't provide new information about data constraints.
• Non-Trivial Functional Dependency: A functional dependency A→B is non-trivial if B is not a subset of A (i.e., B⊆A). These are the functional dependencies that define meaningful constraints and relationships between different attributes in your database schema. They are the focus of normalization.
• Example:
  • StudentID → StudentName (knowing StudentID gives you new information: StudentName)
  • CourseID → CourseTitle (knowing CourseID gives you new information: CourseTitle)
  • InstructorName → InstructorDept (knowing InstructorName gives you new information: InstructorDept)

Given a set of functional dependencies F that are known to hold for a relation R, the closure of F, denoted as F+, is the set of all functional dependencies that can be logically inferred or derived from F. This includes all the FDs in F itself, all trivial FDs, and all additional FDs that can be generated using a set of inference rules called Armstrong's Axioms (which we'll discuss next).

Computing F+ is crucial for:
- Understanding all the implicit constraints that govern your data.
- Determining if a specific FD is implied by a given set of FDs.
- Identifying candidate keys for a relation.
- Checking if a relation is in a particular normal form.

The closure of a set of attributes A with respect to a set of functional dependencies F, denoted as A+, is the set of all attributes that are functionally determined by A. In other words, if you know the values of the attributes in A, you can determine the values of all attributes in A+.

This concept is particularly useful for finding candidate keys. If A+ contains all the attributes of the relation R, then A is a superkey for R. If A is a superkey and no proper subset of A is also a superkey, then A is a candidate key.

Algorithm to Compute A+ (The "Chase Algorithm"):
1. Initialization: Start with A+=A. (The set of attributes you are given.)
2. Iteration: Repeatedly apply the following rule until no new attributes can be added to A+: For each functional dependency X→Y in the given set F: If all attributes in X are already in A+, then add all attributes in Y to A+.

Example: Let's consider a relation R=(A,B,C,D,E) with the following set of functional dependencies F={A→B,BC→D,AD→E}. Let's find the closure of the attribute set {A,C} (i.e., {A,C}+).
- Step 1: Initialization A+={A,C}
- Step 2: Iteration
- FD: A→B Is A (the determinant) a subset of A+? Yes, A⊆{A,C}. So, add B to A+. Now, A+={A,B,C}.
- FD: BC→D Is BC (the determinant) a subset of A+? Yes, BC⊆{A,B,C}. So, add D to A+. Now, A+={A,B,C,D}.
- FD: AD→E Is AD (the determinant) a subset of A+? Yes, AD⊆{A,B,C,D}. So, add E to A+. Now, A+={A,B,C,D,E}.
- Step 3: Termination No new attributes can be added to A+ in the subsequent iterations.

Therefore, the closure of {A,C} is {A,B,C,D,E}. Since {A,C}+ contains all attributes of the relation R, it means that {A,C} is a superkey for R. If no proper subset of {A,C} (i.e., {A} or {C}) can also determine all attributes, then {A,C} is a candidate key.