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proofs by Mind Map: proofs

1. Proofs (i.e. arguments) are designed to "convince" a particular audience.

1.1. In university courses, you are typically expected to convince someone at your own level (E.g. a friend or peer in your course.)

2. Proofs (i.e. arguments) contain hidden assumptions

3. Statements

3.1. Implication (statement)

3.1.1. Usually you get A and B, where A -> B, then will be expected to prove that it's true.

3.1.1.1. Identify A, and

3.1.1.2. Identify B in your statement

4. Proof techniques

4.1. Forward backward method

4.1.1. Backward process

4.1.2. Forward process

4.1.2.1. combining forward statements

5. "Writing proofs" (Final presentation formats)

5.1. Tabular summary <-> what a mathematician might secretly scribble down to solve a problem <-> not what today's mathematicians write up for publication

5.2. Condensed proofs <-> Paragraph format <-> Contemporary standard

5.2.1. As a reader, you are expected to to fill in the omitted steps from your own background knowledge.

5.2.1.1. However, if you can learn strategies to break down, define parts of, and interpret condensed proofs, then you can teach yourself anything in mathematics. This is because all contemporary mathematics is written in condensed proof format.

6. Definitions and terminology

6.1. uses

6.1.1. forwards process

6.1.2. backwards process

6.2. multiple definitions

6.2.1. "equivalent" statements (A,B)

6.3. Working with notation

6.3.1. matching notation

6.3.2. overlapping notation (AVOID)

6.4. Previous knowledge

6.4.1. backwards

6.4.2. forwards

6.5. terminology

6.5.1. proposition

6.5.2. theorem

6.5.3. corollary

6.5.4. axioms

6.5.5. negation (not A, ~A)

6.5.6. truth table

6.5.6.1. proposition

6.5.6.2. converse

6.5.6.3. inverse

6.5.6.4. contrapositive

6.5.7. equivalent