Notes on Proofs and Calculations for Mathematics Exam Preparation

Key Notes on Proof Writing and Calculations in Exercises and Exams

  • Clarity in Argumentation

    • Use complete and precise German sentences when constructing proofs.
    • A sequence of formulas alone does not constitute a proof.
    • Incorrect logic or missing arguments can result in significant point deductions in exams.
    • Example: If using mathematical induction for a claim ( \forall n \in \mathbb{N}: \phi(n) ), and you want to use ( \phi(n-1) ) in the inductive step ( n \to n + 1 ), then the induction hypothesis should include a wider scope, e.g., ( \forall m \in \mathbb{N}, (m \leq n \implies \phi(m)) ).

  • Awareness of Changing Proof Objectives

    • Be mindful of the proof objective at every stage, which may change throughout.
    • Document key points, e.g., “To show: ( \text{claim} )” or “We will now show: ( \text{intermediate claim} )”.
    • Mark the end of the proof clearly, using words like “This was to show” or symbols like “✷” or “q.e.d.” (quod erat demonstrandum).

  • Awareness of Given Statements

    • The list of given statements evolves during the proof.
    • Clearly state which statements are relevant at various proof points. For example, use “Because of ( \text{formula number} ), we know ( \text{statement} )” or “Given ( \text{statement} )”.
    • Critical documentation is necessary at the beginning, end, and at any intermission in the proof where new assumptions are introduced.
    • Introduce new assumptions using phrases like “We assume: ( \text{statement} )” or “Let ( \text{statement} ) hold”. Mark important statements for easy reference later.

  • Awareness of Given Objects

    • The status of the objects (variables) changes throughout the proof.
    • Document the introduction of new variables with phrases like “Let ( ext{Variable} ) with ( ext{type specification} ) be given”.
    • For instance, one may say “Let ( \delta > 0 ) be given” or more succinctly “Let ( \delta > 0 )”.
    • It's vital to bind newly introduced free variables properly, often using quantifiers, e.g., “( \forall x: \phi(x) \text{ for all } x \text{ with } \text{type specification} )” or succinctly “( \forall x \in ext{Type}: \phi(x) )”.

  • Critical Review of Calculations

    • Regularly perform consistency checks on your work, such as substituting simple values (like 0 or 1) to test validity.
    • Visualize results geometrically, if possible, to check consistency with your expectations.
    • Spending time checking for errors is beneficial and may prevent extensive mistakes during exams.
    • Beware of errors that can lead to lengthy pointless calculations yielding zero points, which waste valuable time.

  • Usage of Functions and Relations

    • Do not use functions or relations outside of their defined domains.
    • Common errors include dividing by zero or misapplying relations to non-real complex numbers.

  • Avoid Incorrect Calculation Rules

    • When adding fractions, ensure to find a common denominator through expansion and adding the numerators, not merely adding numerators and denominators directly.
    • Be aware that transcendental functions such as ( ext{sin}, ext{cos}, ext{log}, ext{exp} ) are not linear; for example, ( ext{sin}(x + y)
      eq ext{sin}(x) + ext{sin}(y) ) in general.
    • Do not omit necessary terms in transform formulas, as neglecting inner derivatives can lead to significant beginner errors resulting in zero points in exam questions.