Overall Schemes/Methods/Strategies for Proofs

Sections 1.6-1.7 of our textbook

* direct proof of "P --> Q": assume P, prove Q
plus
* use of definitions
  interpreting them in common mathematical way: "if" actually "means"/"iff"
* transforming n^2 to ...
    Equational Reasoning
* being goal-directed
    backward reasoning vs. forward reasoning
* realizing when done
e.g. page077_Example1
e.g. page077_Example2

* proof by contrapositive of "P --> Q" i.e. "if PHI_1 then PHI_2":
  equivalent to "-Q --> -P" i.e. "if not PHI_2 then not PHI_1"
  this depends on work with propositions which we did earlier
e.g. page078_Example3

* vacuous proof of "P --> Q": have this when P is false
e.g. page078_Example5

* trivial proof of "P --> Q": have this when Q is true
                                without needing to use (assume) P
e.g. page079_Example6

* proof by contradiction of "P --> Q": assume P and NOT Q, derive false
    some similarities to proof by contraposition

* proofs of equivalence of "P <--> Q": prove "P --> Q" and "Q --> P"

* proof of existence by example: to prove "Ex P(x)",
    find value to plug in for x making true

* proof of falseness of "Ax P(x)" by counterexample
    in ProofBuilder, enter the formula proving false with NOT in front