Tools and Techniques in Modal Logic Marcus Kracht II ...

CHAPTER 1
Algebra, Logic and Deduction
1.1. Basic Facts and Structures
In this section we will briefly explain the notation as well as the basic facts and
structures which will more or less be presupposed throughout this book. We will
assume that the reader is familiar with them or is at least willing to grant their truth.
Sets, Functions. We write {x : ϕ(x)} for the set of all objects satisfying ϕ. Given
a set S , ℘(S ) denotes the powerset of S , ♯S the cardinality of S . For functions we
write f : A → B to say that f is a function from A to B, and f : x ↦→ y to say that
f maps (in particular) x to y. The image of x under f is denoted by f (x). We write
f : A ↣ B if f is injective, that is, if f (x) = f (y) implies x = y for all x, y ∈ A;
and we write f : A ↠ B if f is surjective, that is, if for every y ∈ B there is an
x ∈ A such that y = f (x). For a set S ⊆ A, f [S ] := { f (x) : x ∈ S }. We put
f −1 (y) := {x : f (x) = y}. For a set T ⊆ B, f −1 [T] := {x : f (x) ∈ T}. If f : A → B
and g : B → C then g ◦ f : A → C is defined by (g ◦ f )(x) := g( f (x)). The image
of f : A → B, denoted by im[ f ], is defined by im[ f ] := f [A]. M N denotes the set of
functions from N to M. If C ⊆ A then f ↾ C denotes the restriction of f to the set C.
Cardinal and Ordinal Numbers. Finite ordinal numbers are constructed as follows.
We start with the empty set, which is denoted by 0. The number n is the set
{0, 1, . . . , n − 1}. ‘i < n’ is synonymous with ‘i ∈ n’. In general, an ordinal number is
the set of ordinal numbers smaller than that number. So, in constructing ordinals, the
next one is always the set of the previously constructed ordinals. There are two types
of ordinal numbers distinct from 0, successor ordinals and limit ordinals. An ordinal
λ is a successor ordinal if it is of the form κ ∪ {κ}, and a limit ordinal if it is not 0 and
not a successor ordinal. Finite numbers are successor ordinals, with the exception of
0. Ordinal numbers are well–ordered by the inclusion relation ∈. A well–order < on
a set R is a linear ordering which is irreflexive and such that any nonempty subset
S ⊆ R has a least element with respect to