Preliminaries

Be X, Y sets, and P(X) the powerset of X. Then C_X = card(X) = |X| is the number of elements of the set X. Thus we have

• C_∅ = |∅| = 0, C_ℕ = C_ℝ = ∞,
• C_X∩Y = |X∩Y|, C_X∪Y = |X∪Y|, C_X-Y = |X-Y|, C_P(X) = |P(X)| =  2^|x|

and the following axioms hold:

• C_X∪Y = C_X-Y + C_X∩Y =C_Y-X + C_X∩Y
• C_X∩Y = C_X∪Y - C_X-Y - C_Y-X
• C_X-Y = C_X∪Y - C_Y
• C_Y-X = C_X∪Y - C_X

An essential basis for assessing the similarity or equality of knowledge subjects is the set of characteristics that they possess. These can be both the Data Properties (DP) as internal characteristics or the Object Properties (OP) as external characteristics. Be AV the set of all values of Atomic Data Types ADT = {:String, :Integer, :Decimal, :Float, :Boolean, :Date, :Time, :anyURI, ...}. and E the set of all particulars of the KG.

With VS : A ∪ R → AV ∪ E the Value Set (VS) of a property is defined as:

{{definition:VS:Value Set of p:VS(p) = { v | (s, p, v) ∈ KG) }. }}

{{definition:FS:Feature Set of p:FS(p) = { (p, v) | (s, p, v) ∈ KG) }. }}

In the case of data properties p ∈ A, the additional restriction applies that the values v must come from the value set VS(p). Then we define the set of all features of the data property p as:

{{definition:DPFS:Data Property Feature Set of p:DPFS(p) = { (p, v) | (p, v) ∈ FS(p) and v ∈ VS(p) }. }}

Examples:

• FS(.Gender) = {(.Gender, male), (.Gender, female)}
• FS(.PhysicalState) = {(.PhysicalState, solid), (.PhysicalState, fluid), (.PhysicalState, gaseous)}
• FS(»Husband) = {(»Husband, >NPS-Richard_Burton)}
• FS(»DesignerOf) = {(»DesignerOf, ^Lamborghini_Miura)}