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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.

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stillenat
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  • This has nothing to do with relations (or set theory), but much more to due with the interpretation of quantification. These details should all be taught and learned when learning about quantification. – Git Gud Jan 30 '16 at 17:03
  • As far as math notation goes, I would agree that it might have to do with the lack of understanding of quantifiers. However, the above definition that seems to be written in plain English doesn't make it any clearer. – stillenat Jan 30 '16 at 17:11
  • The phrase "for all x, for all y, ..." does not imply or presuppose that $x\ne y$. Similarly, "for all x,y,z, ..." doesn't imply that all three things are distinct. If they're meant to be distinct it would have to say so, by adding that condition: "for all $x, y, z$(if $x\ne y$ and $x\ne z$ and $y\ne z$ then ...)". – BrianO Jan 30 '16 at 23:54

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