This is a collection of all important equations in Competitive Programming.

Motivation

Do you find it bad if you couldn’t solve a problem just because you didn’t know about a certain equation?

Do you find it difficult to take a quick look at an equation that you know exists but forgot about it while solving a problem?

Do you think that all the equations that are important in CP are just cluttered and you are too lazy to collect them in one place?

Well, then you are in the right place! I am here to solve all of the aforementioned problems. I believe you should not waste your precious time searching the internet for important equations. You should solve more problems. I am here to undertake the nasty task of collecting things.

The Equation List

1. Axiom of extensionality: $$$\forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow x=y]$$$
2. Axiom of regularity: $$$\forall x[\exists a(a\in x)\Rightarrow\exists y(y\in x\land\lnot\exists z(z\in y\land z\in x))]$$$
3. Axiom schema of specification: $$$\forall z\forall w_1\forall w_2\dots\forall w_n\exists y\forall x[x\in y\Leftrightarrow((x\in z)\land\phi)]$$$
4. Axiom of pairing: $$$\forall x\forall y\exists z((x\in z)\land(y\in z))$$$
5. Axiom of union: $$$\forall\mathcal F\exists A\forall Y\forall x[(x\in Y\land Y\in\mathcal F)\Rightarrow x\in A]$$$
6. Axiom schema of replacement: $$$\forall A\forall w_1\forall w_2\dots\forall w_n[\forall x(x\in A\Rightarrow\exists!y\phi)\Rightarrow\exists B\forall x(x\in A\Rightarrow\exists y(y\in B\land\phi))]$$$
7. Axiom of infinity: $$$\exists X[\exists e(\forall z\lnot(z\in e))\land e\in X\land \forall y(y\in X\Rightarrow y\cup\{y\}\in X)]$$$
8. Axiom of power set: $$$\forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y]$$$
9. Axiom of choice: $$$\forall X[\emptyset\not\in X\Rightarrow\exists f:X\to\bigcup x(\forall A\in X(f(A)\in A))]$$$

Outro

A good life is just a series of good days. So make sure to have a good day, friend .