Reason for various binary relation definitions I am a mathematics TA for an introductory course to set theory and a student had a question that got me thinking:
Observe the following definition for a binary relation: Let $A$ and $B$ be sets and let $a \in A$ and $b \in B$. A relation R from the set $A$ to the set $B$ is a subset $R$:={(a,b): a $\in$A and b $\in$ B} $\subseteq$ $AxB$. 
Following the definition of a binary relation in some textbooks authors write: an element $x$ is related to an element $y$, if and only if $(x,y)$ $\in$ R.
And in another textbook (I checked out of curiosity) they just simply wrote: the statement  $(a,b) \in R$ is read as “a is related to b” and we write $aRb$. 
my question is: why don’t most authors write the if and only if statement and others don’t? Is saying “we read $(a,b) \in R$ as a is related to b” another way to word the definition as opposed to the if and only if statement? I know this may seem like a silly question, but I just want to be thorough in explaining such an important topic. 
 A: Short answer : " is read as " means " means$_{df}$ " which in turn means " $\iff_{df}$"
Hence , the 2 definitions say the same thing. 

$\bullet$ Sometimes, the verb " means" has to be taken as a simple implication. 
For example : " $x$ is a husband " means " $x$ is married" . Being a husband implies being married. 
But it is not the case that "being married" means "being a husband" : it does not imply being a husband. 
Note : Since the implication does not " work " in both directions , " being married" cannot be the definition of " being a husband" 
$\bullet$ Sometimes " means" has to be taken as an equivalence, a bi-implication  . 
For example " number $x$ is even " means " number $x$ is a multiple of 2 ". 
This is the same as saying : " $x$ is even " $\iff$ " $x$ is a multiple of 2". 
$\bullet$ In the definition of a binary relation you are referring to, " means" or " reads as" is taken in the second sense. 
So saying that " $(a,b)\in R$ reads as a is related to b" 
is the same as saying 
" $(a,b)\in R \iff$ a is related to b ". 
