The three axioms of functional dependency for the relational databases given by Armstrong are:
Reflexivity: This axiom states that every attribute in a set functionally determines itself. This means that if you have a set of attributes A, and you pick any subset of those attributes X, then the values in X will determine the values in X. For example, if you have a relation with the attributes A, B, and C, then the subset {A} will determine the values of A, the subset {B} will determine the values of B, and the subset {C} will determine the values of C.
Augmentation: This axiom states that if X functionally determines Y, and Z is a set of attributes that includes Y, then X functionally determines Z. This means that if you have a set of attributes X that determines another set of attributes Y, then X will also determine any set of attributes that includes Y. For example, if you have a relation with the attributes A, B, C, and D, and you know that A functionally determines B, then you can say that A also functionally determines the set {B, C, D}, because it includes B.
Transitivity: This axiom states that if X functionally determines Y, and Y functionally determines Z, then X functionally determines Z. This means that if you have a set of attributes X that determines another set of attributes Y, and Y in turn determines another set of attributes Z, then X will also determine Z. For example, if you have a relation with the attributes A, B, C, and D, and you know that A functionally determines B and B functionally determines C, then you can say that A also functionally determines C.
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