Mathematical definitions and notation really confuse me. For example, a definition similar to the following can be found in many textbooks and online:
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c.
In terms of set theory, the transitive relation can be defined as:
$∀a,b,c∈ X : (aRb ∧ bRa) ⇒ aRc$
According to this definition, it would seem to any newcomer that there has to be at least three distinct elements in a set for the condition to hold. But it's not the case. For example, if someone sees a relation R = {(1,1)} on the set A = {1}, they would never think of it as anything but reflexive. However, it is also transitive and symmetric because it can be viewed as an equality relation.
Can a more precise definition be given to symmetry and transitivity? Or at least, shouldn't a footnote be included everywhere saying that elements a, b, and c are really just variables that can all refer to a single element?
Also, this answer says that the definition includes the case when a = b ( = c) but I would disagree and say that it's absolutely unintuitive considering the confusion it produces. This is even less intuitive if you think of the fact that sets don't usually contain duplicate elements since they are irrelevant in terms of sets.