This document explores mathematical properties of love between males and females. It shows that the concept of a "love triangle" is a myth unless homosexuality is allowed. It defines love as a binary relation and examines whether properties like reflexivity, symmetry, and transitivity apply. Through proofs, it determines that transitivity does not apply to love relations between genders, and thus the love triangle cannot exist mathematically. It encourages bringing stronger relationship properties like transitivity into one's own love through compromise.

1. Mathmatical Properties of (sexual) Love
Dilawar Singh
Email :dilawar@ee.iitb.ac.in
February 3, 2011
Abstract
In this paper, we explore primitive mathematical properties of of love.
Here we are only concerned with love between male and females. Among
other things, we show that concept of ’love triangle’ is a myth unless one
allows homosexuality.
1 Mathematical Definitions
Let H be the set of all humans. Let M ⊂ H is the set of all lovable males and
let F ⊂ H be the set of all lovable females. It follows that M ∪ F ⊂ H. Sexual
love is a binary relation. Nothing more is known for sure. Sexual love is mostly
depicted in movies and romantic fiction etc. Lets denote it by ♥. A♥B says
that ’sexual love’ exists between A and B. A love triangle is a construction
which has three vertices and sexual love exists between any of its two node.
If we draw vertices as a dot and ’sexual love’ between two vertices as a line, it
will closely resemble the triable which we all are familiar with since school days.
A♥B implies that A and B belongs to different gender i.e. if A ∈ M then B ∈ F
etc. In plain English, homosexuality is not allowed. If love is non-sexual e.g.
love between mother and her child, between math and mathematician, we’ll
denote it by ♣ but we are going to talk about it in this paper.
2 Properties OF ♥
But by any distorted intuition or otherwise, LOVE is an relationship, no matter
how unfathomable. Lets settle on this and try to explore its mathematical
properties. And hope that there is any hope to construct any algebraic structure
under ♥.
Properties 1. A♥A i.e. one loves oneself. It is known as Reflexivity.
This is generally not true. We will assume that this is ’not’ true.
Properties 2. If A♥B then B♥A i.e. if A loves B then B loves A. It is known
as symmetry.
Going by most accounts and life experiences, it is also not always true except
for movies and fiction.
1

2. Properties 3. If A♥B and B♥C then A♥C. It is known as transitivity.
This is ridiculous. We’ll give a proof of it.
Proof. If A♥B means A is male and B is female. B♥C makes C a male. Now
A♥C is not possible unless one allows homosexuality.
Now we state a theorem.
Theorem 1. Sexual love triangle (love triangle) does not exists.
Proof directly follows from 3. For being a love triangle, transitivity must
hold and this is not the case. We give a detailed proof.
Proof. Without a loss of generality, We can assume that A ∈ M . Now if A♥B
is true then B ∈ F . Further if A♥C then, C ∈ F . In this case A ∈ M
while B, C ∈ F . If A ∈ F then, using the same argument it can be proved
that, B, C ∈ M . In either case two person belongs to a same gender. A
contradiction!
Remark 1. A more general result which can be found in many situations is that
if there is a continuous switching between two states then any three consequent
states comprises of two same states. e.g. Take this state machine, in which,
at every event, a odd number changes to even number or even number to odd
number, like, 1,2,3,4,5,6,7,8,9,10 ... Take any three consequent numbers you’ll
get either 2 odd or 2 even numbers.
Remark 2. If above three properties are true then the relation is called equiv-
alence relation. For ♥, none of them are true in most of the cases. But still
that much gives me to conclude that, Since the equivalence relations are way to
strong, one may try to bring these properties into his/her ♥ for forming a nice
relationship. Property 3 (i.e. transitivity) could be seen differently by differ-
ent people, and in my opinion can be relaxed to accommodate your love-partner
reservations about it. Nonetheless, first two properties are must for a strong
and ever lasting relationship. One more interesting point is that property 1 is
person dependent. It takes only you to build it. Property 2 is like It takes two to
tango’ and Property 3 is dependent on social context. All three environments,
personal, bi-personal (or bi-sexual) and social environments are important and
ought to be dealt with.
Then one may ask, why bring mathematics in. By common consent, this
is true that in Math, cheating is not allowed. This is the most honest subject
around known to humanity. Second, its the curiosity which makes a human
different from donkeys.
Exercise 1. Denote binary relation ’hate’ by ♠ and explore its properties. Does
it have stronger mathematical properties? Send your answers by email to this
author.
2