Partial Dependency (Databases)

A FD (functional dependency) that holds in a relation is partial when removing one of the determining attributes gives a FD that holds in the relation. A FD that isn't partial is full.

Eg: Suppose {A,B} → {C} but also {A} → {C}. Then {A,B} → {C} is partial; {C} is partially functionally dependent on {A,B}; {C} is functionally dependent on a part of {A,B} that is not all of it. The consequent partial FD is not {A} → {C}. Whether that is partial depends on (per the definition of partial FD) whether a subset of {A} determines {C}; whether {} → {C}.

Eg: Here's a relation value where that example condition holds. (A FD holds in a relation variable when it holds in every value that can arise.)

A  B  C
1  1  1
1  2  1
2  1  1

The non-trivial FDs that hold: {A,B} determines {C}, {B,C}, {A,C} & {A,B,C}; {A}, {B} & {} also determine {C}. Of those: {A,B} → {C} is partial per {A} → {C}, {B} → {C} & {} → {C}; {A} → {C} & {B} → {C} are partial per {} → {C}; the others are full.

A functional dependency X → Y is a full functional dependency if removal of any attribute A from X means that the dependency does not hold any more; that is, for any attribute A ε X, (X – {A}) does not functionally determine Y. A functional dependency X → Y is a partial dependency if some attribute A ε X can be removed from X and the dependency still holds; that is, for some A ε X, (X – {A}) → Y.

-- Fundamentals OF Database Systems 6th Edition, Ramez Elmasri & Navathe

Notice that whether a FD is full vs partial doesn't depend on CKs (candidate keys), let alone one CK that you might be calling the PK (primary key).

(A definition of 2NF is that every non-CK attribute is fully functionally determined by every CK. Observe that the only CK is {A,B} & the only non-CK attribute C is partially dependent on it so this value is not in 2NF & indeed it is the lossless join of components/projections onto {A,B} & {A,C}, onto {A,B} & {B,C} & onto {A,B} & {C}.)

(Beware that that textbook's definition of "transitive FD" does not define the same sort of thing as the standard definition of "transitive FD".)

Partial dependency means that a nonprime attribute is functionally dependent on part of a candidate key. (A nonprime attribute is an attribute that's not part of any candidate key.)

For example, let's start with R{ABCD}, and the functional dependencies AB->CD and A->C.

The only candidate key for R is AB. C and D are a nonprime attributes. C is functionally dependent on A. A is part of a candidate key. That's a partial dependency.

Partial dependency implies is a situation where a non-prime attribute(An attribute that does not form part of the determinant(Primary key/Candidate key)) is functionally dependent to a portion/part of a primary key/Candidate key.

Partial Dependency is one kind of functional dependency that occur when primary key must be candidate key and non prime attribute are depends on the subset/part of candidates key (more than one primary key).

Try to understand partial dependency relate through example :

Seller(Id, Product, Price)

Candidate Key : Id, Product
Non prime attribute : Price

Price attribute only depends on only Product attribute which is a subset of candidate key, Not the whole candidate key(Id, Product) key . It is called partial dependency.

So we can say that Product->Price is partial dependency.

I hope this explaination gives a more intuitive appeal to dependency than the answers previously given.

Functional Dependency

An analysis of dependency operates on the attribute level, i.e. one or more attribute is determined by another attribute, it comes before the concept of keys. 'The role of a key is based on the concept of determination. 'Determination is the state in which knowing the value of one attribute makes it possible to determine the value of another.' Database Systems 12ed

Functional dependency is when one or more attributes determine one or more attributes. For instance:

Social Security Number -> First Name, Last Name.

However, by definition of functional dependency:

(SSN, First Name) -> Last Name

This is also a valid functional dependency. The determinants (The attribute that which determines another attribution) are called super key.

Full Functional Dependency

Thus, as a subset of functional dependency, there is the concept of full functional dependency, where the bare minimal determinant is considered. We refer those bare minimal determinants collectively as one candidate key (weird linguistic quirk in my opinion, like the concept of vector).

Partial Functional Dependency

However, sometimes one of the attributes in the candidate key is sufficient to determine another attribute(s), BUT not all, in a relation (a table with no rows). That, is when you have a partial functional dependency within a relation.