A proof is just an argument that something is true. While writing a proof, you will make a series of claims; each claim should be
- clearly described (all terms have definitions)
- true (based on the current set of assumptions)
- obviously true based on what has already been stated.
Proofs are how we study the objects we construct in the course.
Here is a table of proof techniques summarizing proof techniques for the basic logical connectives. These techniques will get you most of the way through most of the proofs in this course. There is also a convenient one-page pdf
|Proposition||Symbol||To prove it||To use it||Logical negation|
|P and Q||(P ∧ Q)||prove both P and Q||you may use either P or Q||(¬P) ∨ (¬Q)|
|P or Q||(P ∨ Q)||You may either prove P or prove Q||case analysis||(¬P) ∧ (¬Q)|
|P is false (or "not P")||¬ P||disprove P||contradiction||P|
|if P then Q (or "P implies Q")||P ⇒ Q||assume P, then prove Q||if you know P, conclude Q||P ∧ ¬ Q|
|for all x, P||∀x, P||choose an arbitrary value x||apply to a specific x||∃ x, ¬ P|
|there exists x such that||∃x, P||give a specific x||use an arbitrary x satisfying P||∀ x, ¬ P|
You can think of these as the valid outlines of a valid proof. When writing a 5-paragraph essay, the structure consists of an introductory paragraph, three supporting paragraphs, and a conclusion. Supporting paragraphs have their own structure (made up of sentences, which themselves have a structure...).
Similarly, a proof of a statement like "for all x, P" has an introduction of an arbitrary variable, followed by another proof (this time of P(x)). This proof in turn will have one of the structures in the table above, and so on.
Note that this process is recursive: most of these techniques say "to prove ..., do ... and then prove ...". In most cases, you must repeatedly apply these techniques to build a complete proof.
This category has only the following subcategory.
Pages in category "Proof techniques"
The following 50 pages are in this category, out of 50 total.
- SP19:Lecture 10 Quantifier review
- FA18:Lecture 12 induction
- SP20:Lecture 12 Induction
- FA18:Lecture 13 strong induction and euclidean division
- SP20:Lecture 13 Strong induction and Euclidean division
- SP18:Lecture 2 Sets
- SP18:Lecture 3 Induction
- FA18:Lecture 3 proofs
- SP19:Lecture 3 Set constructions
- SP20:Lecture 3 Set constructions
- SP18:Lecture 4 Logic
- SP20:Lecture 4 prep
- FA19:Lecture 4 Proof techniques
- SP20:Lecture 4 Proof techniques
- FA18:Lecture 4 proofs
- SP18:Lecture 5 Functions
- SP20:Lecture 5 Functions
- Logical negation