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Let $X = \{1, 2, 3, 4\}$ be a set. Let $Y = \{5, 6, 7, 8\}$ be another set. Define relation $R$ as $R = \{(2, 8), (4, 7), (1, 5)\}$.

Then $2 R 8$, $4 R 7$, $1 R 5$.

Here $R$ is both a relation between the components of ordered pairs and the set of ordered pairs related under a certain operation.

Am I understanding it correctly? What's $R$, actually? Why is it used differently in differnt contexts?

Mauro ALLEGRANZA
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DirtySkank
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2 Answers2

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Yes, $R$ is formally defined as a subset of $X \times Y$. However, we use the notation $xRy$ to mean $(x,y) \in R$. The two are strictly equivalent.

JoeyBF
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Because there is a "common sense" tradition according to which a relation is ... a relation.

If you consider the relation "__is father of...", we usually write : "Paul is father of John".

This "tradition" is still used in current mathematical practice when we write : $n < m$ to signify that "$n$ is less than $m$".

Thus, more or less a century ago, the first "symbolization" of relations was :

$xRy$.

Only after the development of modern set-theory, the genial idea of Wiener in 1914 and Kuratowski in 1921 to "encode" the concept of ordered pair in set-theoretic language, permitted the "reduction" of relations to sets, i.e. a relation is a set of ordered pairs.

Thus, we have that : if the relation $R$ is defined on the set $S$, i.e. $R \subseteq S \times S$, then:

$xRy \text { iff } \langle x,y \rangle \in R$.

Mauro ALLEGRANZA
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  • Hi Mauro, quick question on terminology: given $xRy$, could we read this as "$R$ relates $x$ to $y$"? I understand the most common way of reading $xRy$ is "$x$ is $R$-related to $y$", but regardless I would still appreciate your input :-). – Taylor Rendon Jul 14 '22 at 14:44
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    @TaylorRendon - yes, we can. The usual way is "$x$ is less-than $y$" but we may also say that "the less-than relation relates $x$ to $y$". – Mauro ALLEGRANZA Jul 15 '22 at 06:17