N-ary relation as a combination of binary relations

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  • #1
Uke
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Hello,

I am looking for a formal way to represent an n-ary relation as a combination of binary relations and logical connectives.

Suppose we have a set [tex]A[/tex], a set [tex]B = \{b: b\subseteq A^2\}[/tex] of binary relations over [tex]A[/tex], and a set of logical connectives [tex]C = \{\neg, \wedge, \vee\}[/tex].

We define a set of propositional variables [tex]V=\{b(a_i, a_j): b \in B, a_i, a_j \in A\}[/tex]. We denote the set of all well-formed formulas over [tex]V \cup C[/tex] as [tex]F[/tex].

Given a propositional function [tex]f \in F[/tex] and using it as an indicator function, we can define an n-ary relation [tex]R=\{(a_0, a_1, ... , a_n) \in A^n | I(f(a_0, a_1, ... , a_n))=1: f \in F\}[/tex].

Does it make any sense?
 

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  • #2
Uke
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Suppose we have a set [tex]A[/tex], a set [tex]B = \{b: b\subseteq A^2\}[/tex] of binary relations over [tex]A[/tex], and a set of logical connectives [tex]C = \{\neg, \wedge, \vee\}[/tex]. (1)

We define a set of propositional variables [tex]V=\{b(a_i, a_j): b \in B, a_i, a_j \in A\}[/tex]. We denote the set of all well-formed formulas over [tex]V \cup C[/tex] as [tex]F[/tex]. (2)

Given a propositional function [tex]f \in F[/tex] and using it as an indicator function, we can define an n-ary relation [tex]R=\{(a_0, a_1, ... , a_n) \in A^n | I(f(a_0, a_1, ... , a_n))=1: f \in F\}[/tex]. (3)
Ok, since the topic has had 57 views and no replies, I will try to be more specific and share my concerns regarding the above representation.

First, I use [tex]a_i, a_j[/tex] as variables for elements of [tex]A[/tex] in (2), and then [tex]a_0, a_1,...,a_n[/tex] as elements themselves in (3).

Second, [tex]b(a_i, a_j)[/tex] looks like a bad name for a variable.

Third, [tex]f[/tex] is supposed to be a function of propositional variables (true/false), but as input it has elements of [tex]A[/tex]. So either it should not be referred to as a propositional function or the input has to be elements of [tex]V[/tex]. In the former case what would be the correct classification for such a function? In the latter case how to make a transition from [tex]a_0, a_1,...,a_n[/tex] to [tex]v_0, v_1,...,v_n \in V[/tex]?

Finally, do they still use "propositional variable" and "propositional function" in the modern papers? I cannot find a standard for these.

Please, I am new to sets and logic, I desperately need your feedback.
 
  • #3
Hurkyl
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Do you have an example of what you're looking for?


Anyways, note that a ternary relation on A, B, and C is pretty much the same thing as a binary relation on A and BxC. Does that help?
 

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