N-ary relation as a combination of binary relations

In summary, the author is looking for a way to represent an n-ary relation as a combination of binary relations and logical connectives. The author defines a set of propositional variables V=\{b(a_i, a_j): b \in B, a_i, a_j \in A\} and a propositional function f \in F. Using f as an indicator function, they 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\}. The author notes that a
  • #1
Uke
3
0
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 said:
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
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?
 

1. What is an N-ary relation?

An N-ary relation is a mathematical concept that describes the relationship between multiple elements in a set. It involves more than two elements, and can be thought of as a generalization of binary relations.

2. How is an N-ary relation different from a binary relation?

An N-ary relation involves more than two elements, while a binary relation only involves two elements. Additionally, N-ary relations can be seen as a combination of multiple binary relations, whereas binary relations cannot be combined in the same way.

3. What is meant by "combination of binary relations" in N-ary relations?

This refers to the idea that an N-ary relation can be represented as a combination of multiple binary relations. This means that the relationship between multiple elements in a set can be broken down into simpler relationships between pairs of elements.

4. How are N-ary relations represented?

N-ary relations can be represented in various ways, such as through tables, graphs, or matrices. The specific representation used depends on the context and purpose of the relation.

5. What are some real-life examples of N-ary relations?

N-ary relations can be found in various fields, such as mathematics, computer science, and linguistics. Some examples include family trees, database tables, and hierarchical organizational structures.

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