set theory is a system of notation.
what is important is not what a mathematical object is, but what its properties are.
(x,y)=(u,v) if only if x=u and y=v
(x,y)≠(y,x) if x≠y
crystalline concept means that it has strong links that relate its properties in a coherent and inevitable way.
define the properties that a particular formal structure must have and deduce all its other properties by mathematical proof.
| Connective | Quantifier |
| AND | all |
| OR | some |
| XOR | if only if |
T=P∥(¬P)
C=P&(¬P)
| P | C | ¬P⇒C |
| t | f | t |
| f | f | f |
Such a preliminary result,proved before using them in the proof of a theorem,is called a lemma.
A corollary is a result that can be deduced very simply from a theorem.
By self-explanation,you are much more likely to make firmer links in your brain than you would by passively reading one line after another.
The most important defining feature of natural numbers is the possibility of proving theorems using mathematical induction.
Peano Axioms:
- not surjective
- injective
- inductive