Research about Mathematical Proof

Knowledge Sharing:
1. Mathematics is all about proving that certain statements, such as Pythagoras' theorem, are true everywhere and for eternity.
2. Mathematics is perhaps the only field in which absolute certainty is possible, which is why mathematicians hold proofs so dearly.
3. A theorem is a proven statement that was constructed using previously proven statements, such as theorems, or constructed using axioms. Some theorems are very complicated and involved, so we will discuss their different parts.
4. A constructive proof is the most basic kind of proof there is. It is a proof that starts with a hypothesis, and a person uses a series of logical steps and a list of axioms, to arrive at a conclusion.
5. The method of proof by contradiction is to assume that a statement is not true and then to show that that assumption leads to a contradiction.
6. The beauty of induction is that it allows a theorem to be proven true where an infinite number of cases exist without exploring each case individually.
7. Mathematical induction is a mathematical proof technique, most commonly used to establish a given statement for all natural numbers, although it can be used to prove statements about any well-ordered set. It is a form of direct proof, and it is done in two steps. The first step, known as the base case, is to prove the given statement for the first natural number. The second step, known as the inductive step, is to prove that the given statement for any one natural number implies the given statement for the next natural number. From these two steps, mathematical induction is the rule from which we infer that the given statement is established for all natural numbers.
8.  Mathematical induction in this extended sense is closely related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.
9. Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other, the nested images that occur are a form of infinite recursion.
10.  Recursively defined mathematical objects include functions, sets, and especially fractals.
11.  Induction is a way of proving mathematical theorems. Like proof by contradiction or direct proof, this method is used to prove a variety of statements. Simplistic in nature, this method makes use of the fact that if a statement is true for some starting condition, and then it can be shown that the statement is true for a general subsequent condition, then, it is true in general. Formally, if we show that a statement is true for an integer n>4, and then we show that it is true for some integer k+1 if it is true for the integer k(k is greater than or equal to 4), then we have shown that it is true for all integers greater than 4.

Proof:
1. Constructive Proof
2. Proof by Contrapositive
3. Proof by Contradiction
4. Proof by Induction - Mathematical Induction
5. Counterexample
6. Proof by Exhaustion

Process:
1. Proof Idea
2. Proof Steps with simple condition
3. Proof Steps with complex condition

Foundations:
1. Recursions
2. Mathematical Induction

Reasoning:
1. Inductive Reasoning, Deductive Reasoning, etc.
2. Axioms -> Theorems

Questions?
1. What is the difference between Mathematical Induction and Recursion?

References:
http://algs4.cs.princeton.edu/42digraph/
https://en.wikibooks.org/wiki/Mathematical_Proof/Methods_of_Proof/Other_Proof_Types
https://en.wikipedia.org/wiki/Mathematical_induction
https://en.wikipedia.org/wiki/Recursion
https://en.wikipedia.org/wiki/Mathematical_proof#History_and_etymology