I am having a hard time understanding some things dealing with these relations. The five relations we are dealing with are reflexive, symmetric, transitive, irreflexive, and antisymmetric.
$R$ is reflexive is $xRx$ for all $x \in A$.
$R$ is symmetric if $xRy$ implies $yRx$ for all $x, y \in A$.
$R$ is transitive if $xRy$ and $yRz$ implies $xRz$ for all $x, y, z \in A$.
$R$ is irreflexive if $(x, x) \notin R$ for all $x \in A$.
$R$ is antisymmetric is $xRy$ and $yRx$ implies $x = y$ for all $x, y \in A$.
All of them make sense to me besides the last one.
The example in the book says to list all the properties that apply for the given relation: The "has a common national language with" relation on countries.
I am having trouble deciding which ones it has.
To me it makes sense that a country has a common national language with itself, so I think it's reflexive? Also, let's say you have two countries like the USA and England, it makes sense to me that it is symmetric since USA has a common national language with England and, vice verse, England has a common national language with the USA, but for some of the other examples I think it is symmetric when it is not so I am not certain here.
Transitive makes sense so that if country A has a common national language with country B, and country B has a common national language with country C, that country A has a common national language with country C.
Reflexive makes sense up above, but here is where I get confused since it makes sense for it to be irreflexive as well.
How can it not be antisymmetric as well? From my definitions above, aren't they the same thing?