Given a set $A$, the standard definition (at least Wikipedia's one) says that a binary relation $\sim$ on $A$ is an equivalence relation iff:
- $\forall a\in A \quad a\sim a\quad$ (reflexivity);
- $\forall a,b\in A\quad a\sim b\;\implies\; b\sim a\quad$ (symmetry);
- $\forall a,b,c\in A\quad a\sim b$ and $b\sim c\;\implies a\sim c\quad$ (transitivity).
I had been believing for a while that the first assumption (reflexivity) was unnecessary, given the other two. Indeed, if $a\sim b$, then $b\sim a$ by symmetry, and by transitivity $a\sim a$. However, I've then realised that actually these two conditions are not enough. Indeed, taking $A=\mathbb R$ and $a\sim b$ iff $ab>0$ we get that $0$ is not equivalent to $0$, and so the reflexivity does not hold.
The issue in the above example is that in order to use the symmetry and the transitivity to show the reflexivity for an element $a$, we must assume that there exists an element $b\in A$ such that $a\sim b$ or $b\sim a$. This observation leads to the following alternative definition. $\sim$ is an equivalence relation on $A$ iff:
- $\forall a\in A \quad \exists b:a\sim b$ or $b\sim a\quad$ (existence);
- $\forall a,b\in A\quad a\sim b\;\implies\; b\sim a\quad$ (symmetry);
- $\forall a,b,c\in A\quad a\sim b$ and $b\sim c\;\implies a\sim c\quad$ (transitivity).
The two definitions are equivalent, but the single existence condition, without symmetry and transitivity, is weaker than the reflexivity assumption, as the latter implies the former.
Are there other ways to make the assumptions in this definition weaker?
By this I mean, is it possible to replace the existence, the symmetry or the transitivity with a weaker assumption (or possibly a set of weaker assumptions), and still recover the equivalence relation? Alternatively, if this is already a "minimal" definition (in the sense that none of the conditions can be replaced by a weaker one) can we find other weaker definitions of equivalence relation by relaxing the symmetry and transitivity? Is it possible to state all the "minimal" definitions of equivalence relation?
Disclaimer: I am not familiar at all with mathematical logic. I believe this question fits in the logic tag, but if this is not the case please feel free to change it. When I speak about "minimal definition" I intuitively mean that each assumption $\alpha$ in the definition cannot be replaced by a weaker set of assumptions $\{\alpha'_i\}$ (where each $\alpha'_i$ is implied by $\alpha$, but alone does not implies $\alpha$) which still leads to the same definition. Hopefully this makes sense, although I am not sure how to really formalise it.