My doubt arised from the sources below.
Such a class is what we call a binary relation, or just relation.
We will not use the terminology "unary relations"; rather we will just call them classes (or sets, as the case may be)
(Tourlakis, 2011, p. 194) $[1]$
A unary relation on a set x is a subset of x
(Devlin, 1979, p. 13) $[2]$
Question:
The definition of a relation I learnt is: a relation is a set of ordered pairs obtained by the association between two sets, which is a subset of their cartesian product.
Now, assuming a unary relation is a set, then the definition I've stated applies for only binary relations?
In summary, I have two questions:
- Is an unary relation a set?
- Do some authors use the term relation to mean binary relations?
References:
$[1]$: Tourlakis, G. (2011). Lectures in Logic and Set Theory: Volume 2, Set Theory (Cambridge Studies in Advanced Mathematics, Series Number 83) (1st ed.). Cambridge University Press. URL
$[2]$: Devlin, K. (1979). Fundamentals of Contemporary Set Theory. Springer. URL
Best Answer
If $I$ and $X$ are sets, recall that $X^I$ is the set of all functions $f\colon I\to X$. When $I=\{1,2,\ldots,n\}$, we often think of $X^I$ as a set of $n$-tuples; the connection is that the $n$-tuple $(x_1,\ldots,x_n)$ corresponds to the function $f\colon\{1,\ldots,n\}\to X$ given by $f(i)=x_i$. In this case, we say the relation is an $n$-ary relation.
Likewise, we may express an $I$-tuple in the form $(x_i)_{i\in I}$, where $x_i$ is the image of $i\in I$ under the element.
If $I$ and $X$ are sets, an $I$-ary relation on $X$ is a subset $R$ of $X^I$. We say an $I$-tuple $(x_i)_{i\in I}$ is in the relation $R$ if and only if $(x_i)\in R$.
For example, for real numbers, we can define a $3$-ary relation $IB$, "in between", where we say $(x,y,z)\in IB$ if and only if $y$ is strictly between $x$ and $z$ (that is, $x\lt y\lt z$).
Generally speaking, binary (i.e., $2$-ary) relations are among the most common relations, so we use the "naked" term "relation" to refer to binary relations, unless explicitly stated otherwise. The definition you quote is the definition of a binary relation. While we often refer to binary relations by just saying "relation", we almost never refer to other types of relations without specifying the index set.
A unary relation would then be a $1$-ary relation. Technically, that means a subset of $X^1$; but it is easy to see that there is a natural bijection between $X^1$ and $X$, identifying $(a_1)$ with just $a_1\in X$, and hence that $1$-ary relation, subsets of $X^1$, are naturally identifiable with subsets of $X$. So in a sense, unary relations "are the same" as subsets of $X$.