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In the book Advanced Calculus by Shlomo and Sternberg (Chapter 0, Section 6), the inverse of an relation is defined as follows:

"The inverse $ R^{-1} $, of a relation R is the set of ordered pairs obtained by reversing those of R:

$$ R^{-1} = \{\langle x,y\rangle\, |\,\langle y,x\rangle \in R\ \} $$ "

It seems that this definition does not actually reverse all the ordered pairs in R, or am I wrong?

js1
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2 Answers2

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Take, as an example, the relation $R$ on $\mathbb{R}$, defined by $(x,y) \in R$ iff $x \leq y$. That is, $R = \{ (x,y) ~|~ x \leq y \}$, so $(1,2) \in R$, but $(5,2) \notin R$. Then the a pair $(x,y)$ satisfies the inverse relation, i.e. $(x,y) \in R^{-1}$, iff $(y,x) \in R$. In our example this means that $$ (x,y) \in R^{-1} \Leftrightarrow (y,x) \in R \Leftrightarrow y \leq x \Leftrightarrow x \geq y. $$ For example, the element $(5,2)$ which did not satisfy $R$, now satisfies $R^{-1}$ because $(2,5) \in R$. Thus the relation "$\geq$" is the inverse relation of "$\leq$", which makes sense.

So the definition looks about right, don't you agree?

Andre
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  • What about symetric and antisymetric relations. – hamam_Abdallah Jul 27 '17 at 12:50
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    A symmetric relation $S$ would then be characterised by $S = S^{-1}$. The relation in my example is an example of an antisymmetric relation. There $(x,y) \in R$ and $(x,y) \in R^{-1}$ if and only if $x=y$, or to put it in fancier words: "the intersection of $R$ and $R^{-1}$ equals the diagonal". – Andre Jul 27 '17 at 12:53
  • Thanks for your answers. I think I overthinked a little. I used the simple definition that a relation is a set of ordered pairs. – js1 Jul 27 '17 at 18:16
  • Well, that is the definition of a relation, but it is a bit abstract and I think it helps a great deal if you think of examples when considering relations. The "ordered pair" $(x,y)$ thing just means that the first entry of your pair, i.e. $x$, is related to the second entry, $y$. That is why some people like to write $xRy$ instead of $(x,y) \in R$, so you can better think of examples like $x=y$, $x \leq y$, $x \sim y$, $x \equiv y$ etc. :) – Andre Jul 27 '17 at 18:55
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The definition might be easier to understand when it is written like this:

the inverse relation of R = { pairs (y,x) | the pair (x,y) belongs to R }

That is the way it is phrased in Lipschutz, Schaum's Outline of Set Theory ( Chapter 6) ( available at Archive.org).

I think this manner of defining " inverse relation" makes more explicit the fact that the inverse relation of R is simply the set of all the " inverse pairs" of the pairs belonging to R.