Armstrong’s Axioms for Functional Dependency(FD)

Armstrong’s Axioms for Functional Dependency(FD)

  • Armstrong’s Axioms is a set of rules.
  • Armstrong’s Axioms is used to derive new FDs from other FDs.
  • It provides a simple technique for reasoning about functional dependencies.
  • It is used to infer all the functional dependencies on a relational database.

If custid or ano is removed from the primary key access_date cannot be determined uniquely. So access_date is fully functional dependent on custid and ano .

But balance and bname depend only on the ano . If custid is removed from the primary key even then dependency of balance and bname on ano holds.

So balance and bname do not fully functionally

dependent on primary key i.e. custid and ano.In other word balance and bname partially depend on primary key.

Various Axioms Rules

Rule 1:

Reflexivity.

If A is a set of attributes and B is a subset of A, then A holds B. { A -> B }

Rule 2:

Augmentation

If A hold B and C is a set of attributes, then AC holds BC. {AC -> BC}
It means that attribute in dependencies does not change the basic dependencies.

Rule:3

Transitivity

If A holds B and B holds C, then A holds C.
If {A -> B} and {B -> C}, then {A -> C}
A holds B {A -> B} means that A functionally determines B.

Rule:4 Union

If A holds B and A holds C, then A holds BC.
If{A -> B} and {A -> C}, then {A -> BC}

Rule:5 Decomposition

If A holds BC and A holds B, then A holds C.
If{A -> BC} and {A -> B}, then {A -> C}

Rule:6 Composition

If A holds B and X holds Y, then AX holds BY.
If{A -> B} and {X-> Y}, then {AX->BY}

Rule:7 Pseudo Transitivity

If A holds B and BC holds D, then AC holds D.
If{A -> B} and {BC -> D}, then {AC -> D}

Rule:8 Self-determination

A ->A

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