Describe 4 types of Mathematical Proof: Algebraic Proof, Visual Proof, Logic Proof, Algorithmic Proof. Kung Fu Computer Science, Geometric complexity theory

1. 4 Types of Mathematical
Proofs
Sing Kuang Tan
singkuangtan@gmail.com
14 July 2021
Link to my paper
https://www.slideshare.net/SingKuangTan/brief-np-vspexplain-249524831
Prove Np not equal P using Markov Random Field and Boolean Algebra
Simplification
https://vixra.org/abs/2105.0181

2. Types of Mathematical Proofs
• In my opinion, there are 4 types of Mathematical Proofs
• Algebraic Proof
• Visual Proof
• Logic Proof
• Algorithmic Proof

3. Algebraic Proof
• https://www.tuitionmath.com/single-post/2016/12/09/11-tips-to-
conquer-trigonometry-proving

4. Visual Proof
• https://www.maa.org/press/periodicals/convergence/proofs-
without-words-and-beyond-pwws-and-mathematical-proof
• http://www.ams.org/publicoutreach/feature-column/fcarc-visual1
Pythagorean Theorem
a2+b2=c2

5. Logic Proof
• Proof using Mathematical Logic
• E.g.
• p⇒q
• q⇒r
• ∴ p⇒r(3.2.3)
• Proof: No least positive rational number
• https://en.wikipedia.org/wiki/Proof_by_contradiction
• Consider the proposition, P: "there is no smallest rational number greater than 0". In a proof
by contradiction, we start by assuming the opposite, ¬P: that there is a smallest rational
number, say, r.
• Now, r/2 is a rational number greater than 0 and smaller than r. But that contradicts the
assumption that r was the smallest rational number (if "r is the smallest rational number"
were Q, then one can infer from "r/2 is a rational number smaller than r" that ¬Q.) This
contradictions shows that the original proposition, P, must be true. That is, that "there is no
smallest rational number greater than 0".

6. Algorithmic Proof
• Proof the Cycle Property of Minimum Spanning Tree
• https://en.wikipedia.org/wiki/Minimum_spanning_tree#Cycle_property
• Cycle property
• For any cycle C in the graph, if the weight of an edge e of C is larger than the
individual weights of all other edges of C, then this edge cannot belong to an
MST.
• Proof: Assume the contrary, i.e. that e belongs to an MST T1. Then
deleting e will break T1 into two subtrees with the two ends of e in different
subtrees. The remainder of C reconnects the subtrees, hence there is an
edge f of C with ends in different subtrees, i.e., it reconnects the subtrees into
a tree T2 with weight less than that of T1, because the weight of f is less than
the weight of e.

7. • Algorithmic Proof
• Problem Reduction
• are also used to proof a problem e.g. 3 coloring problem is NP problem
• by converting another NP problem e.g. 3sat to the original 3 coloring problem
• Proof correctness of an algorithm
• Proof time and space complexity of an algorithm
• Proof that 2 algorithms are equivalent
• …

8. My NP vs P proof
• My NP vs P proof is proved fully using only algebra
• Without any algorithm
• I think algebraic proof is the most advance form of proving in Mathematical Proof
• 我的NP vs P proof证明是完全用代数来证明
• 没有用到任何算法
• 我认为代数是最先进的数学证明方法
• Read my Paper:
• https://www.slideshare.net/SingKuangTan/brief-np-vspexplain-249524831
• Prove Np not equal P using Markov Random Field and Boolean Algebra Simplification
• https://vixra.org/abs/2105.0181

9. Share my links
• I am a Small Person with Big Dreams
• Please help me to repost my links to other platforms so that I can spread my ideas to the rest
of the world
• 我人小，但因梦想而伟大。
• 请帮我的文件链接传发到其他平台，让我的思想能传遍天下。
• Comments? Send to singkuangtan@gmail.com
• Link to my paper
• https://www.slideshare.net/SingKuangTan/brief-np-vspexplain-249524831
• Prove Np not equal P using Markov Random Field and Boolean Algebra Simplification
• https://vixra.org/abs/2105.0181