N-ary relation as a combination of binary relations

[Uke]

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 A, a set B = \{b: b\subseteq A^2\} of binary relations over A, and a set of logical connectives C = \{\neg, \wedge, \vee\}.

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

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

Does it make any sense?

 

[Uke]

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

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

Given a propositional function f \in F and using it as an indicator function, we can define an n-ary relation R=\{(a_0, a_1, ... , a_n) \in A^n | I(f(a_0, a_1, ... , a_n))=1: f \in F\}. (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 a_i, a_j as variables for elements of A in (2), and then a_0, a_1,...,a_n as elements themselves in (3).

Second, b(a_i, a_j) looks like a bad name for a variable.

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

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.

 

[Hurkyl]

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?