🐣 How to self-explain

🐥 To improve your understanding of a proof, there is a series of techniques you should apply.

• 🦉 After reading each line:
  • 🦚 Try to identify and elaborate the main ideas in the proof.
  • 🪿 Attempt to explain each line in terms of previous ideas. These may be ideas from the information in the proof, ideas from previous theorems/proofs, or ideas from your own prior knowledge of the topic area.
  • 🦢 Consider any questions that arise if new information contradicts your current understanding. Before proceeding to the next line of the proof you should ask yourself the following:

• 🦆 Do I understand the ideas used in that line?
• 🦩 Do I understand why those ideas have been used?
• 🐓 How do those ideas link to other ideas in the proof, other theorems, or prior knowledge that I may have?
• 🦤 Does the self-explanation I have generated help to answer the questions that I am asking? 🐦‍🔥 By engaging in self-explanation, you deepen your understanding and ensure that every step in a proof is clear and justified.

The Philosophy of Proofs

The most important thing about proofs is that they are precise descriptions of the though process for a statement's rationalization.

It is tradition to finish proofs with the latin words: "Quod erat demonstradum", or the abrebiation $QED$

🌬️ Induction & Proof Techniques

💧 Mathematical Induction

• 💧 Prove formulas for sums, or recursion patterns.

☔️ Proof by Contradiction, Construction, etc.

• ☔️ “A rope-and-nails” style logic puzzle, “infinite descent.”

Proofs

Prove only the things that are true

A proof is a valid argument that establishes the truth of a mathematical statement.

A proof can use the hypotheses of the theorem, if any, axioms assumed to be true, and previously proven theorems.

Using these ingredients and rules of inference, the final step of the proof establishes the truth of the statement being proved.

A theorem is a statement that can be shown to be true

Less important theorems sometimes are called propositions.

Theorems can also be referred to as facts or results.

We demonstrate that a theorem is true with a proof.

A proof is a valid argument that establishes the truth of a theorem.

The statements used in a proof can include axioms (or postulates), which are statements we assume to be true.

A less important theorem that is helpful in the proof of other results is called a lemma (plural lemmas or lemmata).

A corollary is a theorem that can be established directly from a theorem that has been proved.

A conjecture is a statement that is being proposed to be a true statement, usually on the basis of some partial evidence, a heuristic argument, or the intuition of an expert.

When a proof of a conjecture is found, the conjecture becomes a theorem.

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📍 DIRECT PROOF:

The square of an odd integer is odd.

1. Let $𝚗∈ℕ$
2. Then $n$ is 1 more than an even integer, so $\color{gold}n = 1 + 2m$ for some integer $m$.
   1. $n^2$ = $(1+2m)^2$
   2. $n^2$ = $(1+2m)(1+2m)$ = $1 + 4m +4m^2$ = $1 + 4(m+m^2)$
3. This is 1 more than $\color{orange}4(m + m^2)$, an even number.
   $\color{gold}∴ n^2 \text{ is odd.}$

\color{gold}
∎ n: \text{ odd} ⟹ n^2: \text{ odd, } n^2 = 1+4k \;\;(k∈ℕ)

This is a $\color{gold}DIRECT$ proof.

The square of a multiple of 3 is also multiple of 3

1. Let $𝚗∈ℕ$
2. Then $n∷3$, so $\color{gold}n = 3m$ for some integer $m$.
   1. $n^2$ = $(3m)^2$
   2. $n^2$ = $9·m^2$
3. 9 is multiple of 3. $3∣9$
   $\color{gold}∴ n^2 \text{ is multiple of 3.}$

\color{gold}
∎ n: 3 ∣ n ⟹ n^2: \left\{\begin{matrix} 3 ∣ n^2, \\ 9 ∣ n^2 \end{matrix}\right.

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PROOF BY CONTRADICTION

No real number has square equal to −1

1. Suppose the statement is false. This means that there is a real number, say $x$, such that $\color{gold}x: x∈ℝ, x^2 =−1$.
2. A property of real numbers is that $\color{lime}∀r∊ℝ : r^2 ≥ 0$
3. $x$ is a real number, so $x:$ $\color{gold}x^2 = -1$ AND $\color{lime}x^2≥ 0$
4. 3 Is a contradiction. 1 must be false
5. ∴ No real number has square equal to −1

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