I feel like this questions should be really simple, so I am probably not grasping the concept of binary relation correctly.
This is an example of a binary relation that is symmetric and transitive but not reflexive, stated in Dugundji's book "Topology".
His intuitive definition of a binary relation $R$ in a set $A$ is:
A proposition such that for each ordered couple $(a,b)$ of elements of $A$, one can determine whether $aRb$ ('$a$ is in relation $R$ to $b$') is or is not true."
His formal definition is:
A binary relation $R$ in a set $A$ is a subset $R \subset A \times A$. $(a,b) \in R$ is written $aRb$.
So a binary relation $B \times B$ on $A$ is simply a pair $(a,b)$ such that $(a,b) \in B \times B$, or $a \in B$ AND $b \in B$, correct?
Then, on the example above, the relation is any pair of points that belongs to $[0,1] \times [0,1]$. If reflexivity is defined as $\forall a \in A : aRa$, why is this relation not reflexive, if for any pair $(a,a)$, it either belongs or not belongs to $[0,1] \times [0,1]$?
Thank you!
The relation $R$ is defined by $aRb$ if and only if $(a,b)\in[0,1]\times[0,1]$.
If $a\notin[0,1]$, then $(a,a)\notin[0,1]\times[0,1]$, so it's not true that $aRa$, and so the relation is not reflexive.