# 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?

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#### 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

Staff Emeritus
Gold Member
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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