Propositional Logic
Grammar a, b, c.. | T | F | A  B | A  B | A
Model
A  UM , T  UM , F  0
M
Validity
M
M
A valid in M  A M  U M   A
A entails B  A M  B
M
 AB
A  B   (A  B)
 A  TA A,  A  A is a tautology
Functions
Function x, y,y'.(x, y)  f(x)  (x, y')  f(x)  y  y'
Total  x  X . f ( x) is defined
Injective x, x' X . f ( x)  f ( x' )  x  x'
Surjective
y  Y .x  X . y  f ( x)
Bijective
Images
Bijective  has an inverse
RA  { y  Y | x  A.( x, y )  R}
R 1 A  {x  X | y  A.( x, y)  R}
Relations
Equivalence
Partial Order
Lub
Glb
Set Size
Definition
Countable
x  X .xRx , x, y  X .xRy  yRx
x, y, z  X .xRy  yRz  xRz
Class = {x}R  { y  X | yRx}
p  P. p  p
p, q  P. p  q  q  p  q  p
p, q, r  P. p  q  q  r  p  r
Total  all pairs comparable
A preorder lacks anti-symmetry
X  P: u such that x  X .x  u
p  P.(x  X .x  p)  u  p)
X  P: l such that x  X .x  l
p  P.(x  X .x  p)  l  p)
If a partial order has all lub, glb
it is called a complete lattice
Indexed Set
Disjoint Union
Miscellaneous
Foundation
Characteristic
Size
Inductive Definition
Rules
Premise (X), axiom (y): X/y
R-closure  ( X / y ). X  Q  y  Q
Bounded  exists R-closed set
I R  {Q | Q is R - closed}
Induction
Transitive
Closure
( X / y)  R.(x  X .x  I R  P( x))  P( y)
Derivation
y  I R  R - derivation of y
Induction can occur on derivations
Finite  f . f : {m   | m  n}  A
A finite
A infinite  f . f :   A
f , B  . f bij : B  A
f . f inj : A  
f , B countable . f inj : A  B
is countable
Union of countable sets countable
 is uncountabl e
Set Construction
S  {x | x  x} . Is S  S?
Paradox
Comprehension {x  X | P ( x)}
P( X )  {Y | Y  X }
Powerset
R   {( a, b) | (a.b)  R  ((a, c)  R   (c, b)  R  )}
R *  R   id R
Well Founded Induction
Definition Relation  such that there are no
infinite descending chains of 
a b a  ba b
Relation
 on A well founded 
Q  A.(m  Q  b  m.b  Q)
To Sets
a  A.((b  a.P(b))  P(a))
Induction
Building
Transitive closure retains w.f.
Product of two w.f.
Lexicographic product of two w.f
f : A  B,  B  (a  A a'  f (a)  B f (a' ))
Recursion
A  B  A  B  A,B in bij. cor.
{xi | i  I }
({1}  A}  ({2}  B)
Union, intersection, product,
set difference
 is well founded
X Y : X  {T , F}  (T  x  Y )
Sets are never in bijective
correspondence with their
powerset: this means there is
no largest set
yields w.f. A where B is w.f.
Given B,  on B well founded and
a F mapping B and an arbitrary #
of C to C for all inputs, we have:
f (b)  F (b, f (b'1 ),.. f (bn ),..), b1  ..  bn  ..
Where f is a unique total function