Is Love $\subseteq$ Person $\times$ Person an equivalence relation, partial order or total order?

Question

Is the predicate Love $\subseteq$ Person $\times$ Person a equivalence relation, partial order or total order? Love $\subseteq$ Person $\times$ Person is valid if $x$ loves $y.$

My conclusion so far: Love $\subseteq$ Person $x$ Person is not reflexive or irreflexive because we don't know if the person loves himself. It can be a partial order as it contains minimal and maximal elements (in case $x$ do or don't love $y$).

It's hard to say if it's transitive because theres only two elements.

Can anyone help me out and describe why/why not it's an equivalence relation, partial order or total/linear order?

Answer 1

If we supposed that Love was a partial order then we immediately run into issues with anti-symmetry. To be precise; if $x$ loves $y$ and $y$ loves $x$ we have the equations $x \leq y$ and $y \leq x$, thus by the anti-symmetry axiom we have $x=y$ and so $x$ and $y$ are the same person. Assuming that you don't take the narcissistic view that people only love themselves I would say this does not hold always. A total order would also run into the same problem.

As for it being an equivalence relation I would say no as it fails on all 3 criteria:

• Reflexivity: This assumes that everyone loves themselves, something I would say is unfortunately not true for everyone
• Symmetry: This says that if you love someone then that person must love you back. If this was true life would indeed be more simple.
• Transitivity: This states that if $a$ loves $b$ and $b$ loves $c$ then $a$ loves $c$ also. If this was true then love triangles wouldn't be so much of a big deal and Hollywood would have an even smaller list of plot lines to draw from.

If I had to categorise I would say your best bet is to think of Love being a directed graph with People as the vertices and an edge from $a$ to $b$ representing $a$ loves $b$.