# A Mathematical Proverb

At any time $t_0$, let $f(a,t_0) = \{a_1, a_2, \ldots, a_n\}$, where each $a_i$ is distinct.

If some $a_i = b$ such that $f(b,t_0) = \{b_1, b_2, \ldots, b_m\}$,

Then all $b_i \in f(b,t_0)$ are said to be friends of $a$.

What does the function $f(x,t)$ represent?

and What is this ancient proverb?

• Some of the phrasing seems a bit odd, so I'll just state my understanding and you can tell me if it's wrong. Basically, f(a,t0) gives you a set A of size n. If we put any element of A into f with t0, we get another set B, and everything in B is a friend of a? – Patrick N Aug 19 '15 at 18:59
• That is correct – Roland Aug 19 '15 at 19:26

The function $f(x,t)$:

Enemies of $x$ at time $t$

The proverb is:

The enemy of my enemy is my friend

Explanation:

All $b_i$ are friends of $a$. $b \in b_i$ can be defined in terms of $a$ as $b \in f(f(a,t),t)$ i.e $f^2(a,t)$. This is defined as friend, so we can conclude f = enemy.

• This seems reasonable, and it's what I thought at first blush, but what's the purpose of t? – Patrick N Aug 19 '15 at 19:05
• @PatrickN Haha exactly, I think it's either a Red Herring or just signifies the fact that enemies can change over time – Cain Aug 19 '15 at 19:06
• That would make the formula so narrow, though! I mean, the ancestral enemies of my enemies are still going to be my friends, I would think. What would make sense is for the friends of a to include all bi in f(b,t) for a range of t, {t0-e^-w/H,t0+e^-w/H}, where e represents the degree of initial animosity, decaying at a rate determined by w, the length of time a and ai have been at war, divided by H, the universal hate constant... – Patrick N Aug 19 '15 at 19:15
• @PatrickN I like the way you think. But to keep it simpler, I think we can note that the saying kind of implies that they would not normally be a friend, it's specifically the fact that you share an enemy that creates the friendship. Thus, when you no longer share an enemy, you are probably no longer friends. Think US and Russia from WW2 to Cold War – Cain Aug 19 '15 at 19:19
• Lol. I knew there'd be debate over time, but it seems everyone at least understands why I included it. – Roland Aug 19 '15 at 19:30

The function represents:

Friends of $a$ at time $t$

The proverb is:

The friends of our friends are our friends.

More explained:

The first line defines the set of $a_i$ as friends of $a$ at any time $t$. The second line defines all $b_1$ as friends of $b$ at time $t$ and says $b$ is in the set of friends of $a$. The third line concludes that all $b_i$ are friends of $a$. This math is simplified to: All $b_i$ are friends of $a$ if some friend of $a$, $a_i = b$

• Where does $t$ come into this? – Rohcana Aug 19 '15 at 19:00
• If the proverb went, "People who are friends of my friends at the same time that my friends are my friends are my friends", this might be accurate. – Patrick N Aug 19 '15 at 19:02
• I consider $t$ as an arbitrary time that's only purpose is to keep the idea that the set of friends can change over time. Edited to reflect this. – MisterEman22 Aug 19 '15 at 19:07
• I'm confused as to why someone downvoted my answer, as it is virtually the same as the accepted answer which received two upvotes :/ – MisterEman22 Aug 19 '15 at 19:32
• Also confused. I gave you an up-vote to even it out, but the proverb you answered isn't nearly as well known (the enemy one even has a wiki page), which could be the issue. – Roland Aug 19 '15 at 19:44