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In functional dependency Armstrong inference rules refers to

  1. Reflexive, Augmentation and Decomposition
  2. Transitive, Augmentation and Reflexive
  3. Augmentation, Transitive, Reflexive and Decomposition
  4. Reflexive, Transitive and Decomposition

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Answer: Transitive, Augmentation and Reflexive

Explanation: We can derive additional functional dependencies from the initial set using inference rules. Armstrongs's Axioms are most basic inference rules. These are 3 rules:

  • Reflexivity: If Y is a subset of X, then X → Y
  • Augmentation: If X → Y, then XZ → YZ
  • Transitivity: If X → Y and Y → Z, then X → Z

Additionally there are secondary rules derived from above 3 rules:

  • Union: If X → Y and X → Z, then X → YZ
  • Decomposition: If X → YZ, then X → Y and X → Z
  • Pseudotransitivity: If X → Y and WY → Z, then WX → Z
  • Composition: If X → Y and Z → W, then XZ → YW

Reference: https://en.wikipedia.org/wiki/Functional_dependency

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Inference Rules



  • An inference rule is an assertion that we can apply to a set of functional dependencies to derive other functional dependencies
  • An initial set of functional dependencies is given

    (E.g., determined by some exploratory process)

  • We can derive additional functional dependencies from the initial set using inference rules
  • Armstrongs's Axioms: the most basic inference rules

    • The Armstrong's axioms are the most basic inference rules
    • Let XY and Z be three sets of attributes in the same relation R.
    • The Armstrong's axioms are:
      1. Reflexivity rule: if Y  X then X  Y
      2. Augmentation rule: if X  Y then XZ  YZ
      3. Transitivity rule: if X  Y and Y  Z then X  Z
  • Easier than it looks...
    • The Armstrong's axioms are very intuitive

      I use a specific relation to illustrate them first (before proving their correctness)

    • Consider the relation:
      
         Employee-Department
      
            SSN            fname       lname     DNO        DName
         +-------------+----------+----------+----------+----------+
         | 111-11-1111 | John     | Smith    |    5     | Research |
         +-------------+----------+----------+----------+----------+
         | 222-22-2222 | Jane     | Doe      |    4     | Payroll  |
         +-------------+----------+----------+----------+----------+
         | 333-33-3333 | Pete     | Pan      |    5     | Research |
         +-------------+----------+----------+----------+----------+        
      

    • Examples of Armostrong axioms:
      1. Reflexivity rule: if Y  X then X  Y
         {fname, lname}  {fname}
        
          What it says is: if I see that same values for {fname, lname}
          I must also see that same value for {fname} - kinda obvious :-)    
        

        • Augmentation rule: if X  Y then XZ  YZ
           If {SSN}  {fname}   then:  {SSN, DName}  {fname, DName}
          

          • Transitivity rule: if X  Y and Y  Z then X  Z
            If:
            
               {SSN}  {DNO} 
               {DNO}  {DName}                      
            Then also:
            
               {SSN}  {DName}
            
          • Visit this link below for more information on the same. :)

            http://www.mathcs.emory.edu/~cheung/Courses/377/Syllabus/9-NormalForms/InferenceRules.htm

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