How can an antisymmetric relation be not reflexive?

Question

Reading a book (I do not know if I can mention its title) I found these definition (the following is exactly the quotation from the pages of the book): "For a binary relation R on a set Y, that is, R⊆YxY, it is customary to write xRy rather than (x,y)∈R. A binary relation R on a set X is: - reflexive if xRx; - antisymmetric if xRy and yRx imply x=y. An antisymmetric relation may or may not be reflexive" I do not get how an antisymmetric relation could not be reflexive. Can you explain it conceptually? A concrete example aside the theory would be appreciate. Thanks in advance

Answer 1

The condition for antisymmetry only says that if $xRy$ and $yRx$ are both true, then $x=y$. Put differently, it says that if $x \neq y$, then it's impossible for both $xRy$ and $yRx$ to be true. This doesn't say anything about whether $xRx$ is true or false for some or all $x$.

Conceptually, think of antisymmetry as ensuring that two distinct elements cannot both be $R$ to each other, but it is completely indifferent to whether or not an element is $R$ with itself.

(By the way "reflexive" means that $xRx$ holds for all $x$ in the set. The quote in the question leaves the variable $x$ unquantified in that definition, which is a serious error.)

The relations $<$ and $\leq$ on natural numbers are both antisymmetric, but only the second one is reflexive: Every number $n$ satisfies $n \leq n$, but no number satisfies $n < n$. Antisymmetry holds for both of them because for two distinct numbers $m,n$, only one of $m \leq n$ and $n \leq m$ can be true, and also only one of $m < n$ and $n < m$.