Proof
A proof is a method of communicating a mathematical truth. It is a series of convincing arguments that leaves no doubt that a given proposition is true.
TODO: How to prove algorithms?
Some Definitions
Proposition: Mathematical claim posed in the form of a statement that either needs to be proven or demonstrated false by a valid argument. Theorem: A particularly significant true proposition. Lemma: Subsidiary proposition that is used in the proof of a theorem Corollary: Proposition that follows from a theorem
3.1. Proving Universally Quantified Statements
Proof
To prove the universally quantified statement ”$∀x∈S,P(x)$“:
Choose a representative mathematical object $x∈S$. This cannot be a specific object. It has to be a placeholder, that is, a variable, so that our argument would work for any specific member of the domain $S$.
Then, show that the open sentence $P$ must be true for our representative $x$, using known facts about the elements of $S$.
Disproof
To disprove the universally quantified statement “$∀x∈S,P(x)$”:
Show that $(∃x∈S,¬P(x))$ is true. In other words, find an element $x∈S$ for which the open sentence $P(x)$ is false. This process is called finding a counterexample.
3.2 Proving Existentially Quantified Statements
Proof
To prove the existentially quantified statement “$∃x∈S,P(x)$”:
Provide an explicit value of x from the domain $S$, and show that $P(x)$ is true for this value of $x$. In other words, find an element of $S$ that satisfies property $P$.
Disproof
To disprove the existentially quantified statement “$∃x∈S,P(x)$”:
Show that the universally quantified statement $(∀x∈S,¬P(x))$ is true.
3.3 Proving Implications

To prove the implication “$A⟹B$”, assume that the hypothesis $A$ is true, and use this assumption to show that the conclusion $B$ is true. The hypothesis $A$ is what you start with. The conclusion $B$ is where you must end up.

To prove the universally quantified implication “$∀x∈S,P(x)⟹Q(x)$”: Let $x$ be an arbitrary element of $S$, assume that the hypothesis $P(x)$ is true, and use this assumption to show that the conclusion $Q(x)$ is true.
3.5 Proof by Contrapositive
 If we want to prove that $A⟹B$, we can prove that $(¬B)⟹(¬A)$
 To prove $(∀x∈S,P(x)⟹Q(x))$, prove $(∀x∈S,(¬Q(x))⟹(¬P(x)))$
3.6 Proof by Contradiction
Let A be a statement. Note that either $A$ or $¬A$ must be false, so the compound statement $A∧(¬A)$ is always false. The statement “$A∧(¬A)$ is true” is called a contradiction.
Ex: Prove that $2 $ is irrational.
Prove if and only if
For proving an if and only if statement:

To prove the statement $A⟺B$, it is equivalent to prove both the implication $A⟹B$ and its converse $B⟹A$

To prove the universally quantified statement $(∀x∈S,P(x)⟺Q(x))$, it is equivalent to do either a or b below
$(a)$ Let $x$ be an arbitrary element of $S$. and prove both the implication ”$P(x)⟹Q(x)$” and its converse ”$Q(x)⟹P(x)$” $(b)$ Prove both the universally quantified implication ”$∀x∈S,Q(x)⟹P(x)$” and ”$∀x∈S,P(x)⟹Q(x)$”
Prove Subset
see Set Theory
Other proof Methods:
 Proof by Induction