Relation

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The notion of relations (also known as predicates) is one of the most important fundamental concepts of set theory. The most common of these are the binary relations, so we begin with them. Once they have been established, we generalize to $n$-ary relations, which we apparently don't come across often, but which occur implicitly very frequently in mathematics.


Binary Relations

A binary relation $\mathfrak{R}$ between a space $A$ and a space $B$ is formally defined as a subset of $A \times B$. If $a \in A$ and $b \in B$, we say $a$ is related to $b$ under $\mathfrak{R}$, and write $\mathfrak{R}(a,b)$, or, more commonly, $a\mathfrak{R}b$, iff $(a,b) \in \mathfrak{R}$.

For a more detailed treatment, see Binary relation.


n-ary Relations

An $n$-ary relation $\mathfrak{R}$ over the sets $A_1,...,A_n$ is a subset of $\prod^n_{i=1}A_i$. If for $i=1,...,n, a_i \in A_i$, we say $a_1, ..., a_n$ are related under $\mathfrak{R}$, and write $\mathfrak{R}(a_1,...,a_n)$ (unfortunately though, the other short hand breaks down here) iff $(a_1,...,a_n) \in \mathfrak{R}$. If $A_1 = ... = A_n = A$, we say $\mathfrak{R}$ is an n-ary relation over $A$.

A very common example of an $n$-ary relation is a linear constraint over a vector space $\mathbb{F}^n$ for some field $\mathbb{F}$: $\sum^n_{i=1}c_ix_i = c$, where $(x_1,...,x_n)$ is an element of the vector space and $c_1,...,c_n,c$ are scalars.

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