Armstrong's axioms

Armstrong's axioms are a set of axioms (or, more precisely, inference rules) used to infer all the functional dependencies on a relational database. They were developed by William W. Armstrong in his 1974 paper.The axioms are sound in generating only functional dependencies in the closure of a set of functional dependencies (denoted as ${\displaystyle F^{+}}$) when applied to that set (denoted as ${\displaystyle F}$). They are also complete in that repeated application of these rules will generate all functional dependencies in the closure ${\displaystyle F^{+}}$.

Axioms 

Let ${\displaystyle R(U)}$ be a relation scheme over the set of attributes ${\displaystyle U}$.  ${\displaystyle X}$, ${\displaystyle Y}$, ${\displaystyle Z}$ denotes any subset of ${\displaystyle U}$ and, for short, the union of two sets of attributes ${\displaystyle X}$ and ${\displaystyle Y}$ denoted by ${\displaystyle XY}$ instead of the usual ${\displaystyle X\cup Y}$; this notation is rather standard in database theory when dealing with sets of attributes.

Axiom of reflexivity

If ${\displaystyle X}$ is a set of attributes and ${\displaystyle Y}$ is a subset of ${\displaystyle X}$, then ${\displaystyle X}$ holds ${\displaystyle Y}$. Hereby, ${\displaystyle X}$ holds ${\displaystyle Y}$ [${\displaystyle X\to Y}$] means that ${\displaystyle X}$ functionally determines ${\displaystyle Y}$.

If ${\displaystyle Y\subseteq X}$ then ${\displaystyle X\to Y}$.

Axiom of augmentation

If ${\displaystyle X}$ holds ${\displaystyle Y}$ and ${\displaystyle Z}$ is a set of attributes, then ${\displaystyle XZ}$ holds ${\displaystyle YZ}$. It means that attribute in dependencies does not change the basic dependencies.

If ${\displaystyle X\to Y}$, then ${\displaystyle XZ\to YZ}$ for any ${\displaystyle Z}$.

Axiom of transitivity

If ${\displaystyle X}$ holds ${\displaystyle Y}$ and ${\displaystyle Y}$ holds ${\displaystyle Z}$, then ${\displaystyle X}$ holds ${\displaystyle Z}$.

If ${\displaystyle X\to Y}$ and ${\displaystyle Y\to Z}$, then ${\displaystyle X\to Z}$.

Reference: https://en.wikipedia.org/wiki/Armstrong%27s_axioms