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In this blog, we explore primitive mathematical properties of of love. Here we are only concerned with love between male and females. Among other things, we shall show that concept of ‘love triangle’ is a myth unless one allows homosexuality.MATHEMATICAL PRELIMINARIESLet $latex \mathbb{H}$ be the set of all humans. Let $latex \mathbb{M} \subset \mathbb{H}$ is the set of all lovable Males (definition?) and let  $latex \mathbb{F} \subset \mathbb{H}$ be the set of all lovable females . It follows that  $latex \mathbb{M} \cup \mathbb{F} \subset \mathbb{H}$. 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 $latex \heartsuit$. $latex A \heartsuit B$ says that ‘sexual love’ exists between $latex A$ and $latex B$. 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. $latex A \heartsuit B$ implies that $latex A$ and $latex B$ belongs to different gender i.e. if $latex A \in \mathbb{M}$ then $latex B \in \mathbb{F}$ 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 $\clubsuit$ but we are NOT going to talk about it in this blog since it is written on Valentine day. PROPERTIES OF $latex \heartsuit$ But by any distorted intuition or otherwise, LOVE (whether sexual or not) can be considered a ‘weak form’ of an relationshi. Lets settle on this and try to explore its mathematical properties. And hope that there is any hope to construct any algebraic structure under $latex \heartsuit$. Property 1: $latex A \heartsuit A$ i.e. one loves oneself. It is known as Reflexivity.

This is generally not true. We will assume that this is ‘not’ true.

Property 2: If $latex A\heartsuit B$ then $latex B \heartsuit A$ 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.

Property 3: If $latex A \heartsuit B$ and $latex B \heartsuit C$ then $latex A \heartsuit C$. It is known as transitivity.

This is ridiculous. We’ll give a proof of it.

Proof :

If $latex A \heartsuit B$ means $latex A$ is male and $latex B$ is female. $latex B \heartsuit C$ makes $latex C$ a male. Now $latex A \heartsuit C$ is not possible unless one allows homosexuality since both $latex A$ and $latex C$ are males.

Now we can state a theorem. Theorem :   Sexual love triangle (love triangle) does not exists. Proof :

Proof directly follows from Property 3. For being a love triangle, transitivity must hold and this is not the case. Without a loss of generality, We can assume that $latex A \in M$. Now if $latex A \heartsuit B$ is true then $latex B \in F$. Further if $latex A \heartsuit C$ then, $latex C \in F$. In this case $latex A \in M$ while $latex B \in F$, $latex C \in F$. If $latex A \in F$ then, using the same argument it can be proved that, $latex B, C \in M$. In either case two person belongs to a same gender. A contradiction!

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. If above three properties are true then the relation is called equivalence relation. For $latex \heartsuit$, 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 $latex \heartsuit$ for forming a nice relationship. Property 3 (i.e. transitivity) could be seen differently by different 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. If you have no girl/boy to kiss of today (Valentine day), you would like to do this exercise. Denote binary relation ‘hate’  by $latex \spadesuit$ and explore its properties. Does it have stronger mathematical properties? Does Hate Triangle exist? Send your answers by email to this author. Also prove/disprove that, IITians, Times of India and United States form a love triangle under ‘nonsexual love’. Email me/comment your solutions. -- Dilawar