binary relations 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? 
 A: An example of an antisymmetric relation would be ordering on numbers. $x \le y$ and $y \le x$ imply $x=y$, for any $x$ and $y$.
$\le$ is also reflexive ($x\le x$ for all $x$) and transitive (if $x\le y$ and $y \le z$, then $x \le z$).
A relation which has these three properties (reflexive, transitive, antisymmetric) is called a partial order. (If for every $x, y \in A$ either $x\le y$ or $y \le x$, then we have a total order).
The strict order $\lt$ differs from $\le$ by being irreflexive (it is never the case that $x \lt x$) and asymmetric (not to be confused with antisymmetric; you cannot have both $x\lt y$ and $y \lt x$ at the same time, unlike $\le$.)
Equality is a symmetric relation: $y=x$ implies $x=y$. An equivalencce relation is one which, like $=$, is reflexive, transitive, and symmetric.

Regarding your example, the "common language" relation on countries is 


*

*reflexive (all countries share a common language with themselves);

*not transitive, as illustrated by TravisJ's comment;

*not antisymmetric. A counterexample would be the US and the UK. Both share a common language (English), but they are obviously not the same country.

*In fact, the relation is actually symmetric, since if A shares a language with B, then B also shares a language with A.

A: On numbers define $xRy$ to mean any number dividing $x$ also divides $y$. Now what is  the consequence when $aRb$ and $bRa$ are both true?
