General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Definition 82 We say that a set A is countably infinite, iff there is a bijective function
f : A → N.
Theorem 83 We have the following identities for finite sets A and B
#({a, b, c}) = 3 (e.g. choose f = {〈a, 0〉, 〈b, 1〉, 〈c, 2〉})
#(A ∪ B) ≤ #(A) + #(B)
#(A ∩ B) ≤ min(#(A), #(B))
#(A × B) = #(A) · #(B)
With the definition above, we can prove them (last three Homework)
c○: Michael Kohlhase 56
Next we turn to a higher-order function in the wild. The composition function takes two functions
as arguments and yields a function as a result.
Operations on Functions
Definition 84 If f ∈ A → B and g ∈ B → C are functions, then we call
g ◦ f : A → C; x ↦→ g(f(x))
the composition of g and f (read g “after” f).
Definition 85 Let f ∈ A → B and C ⊆ A, then we call the relation {〈c, b〉 |
c ∈ C ∧ 〈c, b〉 ∈ f} the restriction of f to C.
Definition 86 Let f : A → B be a function, A ′ ⊆ A and B ′ ⊆ B, then we
call f(A ′ ) := {b ∈ B | ∃a ∈ A ′ .〈a, b〉 ∈ f} the image of A ′ under f and f −1 (B ′ ) :=
{a ∈ A | ∃b ∈ B ′ .〈a, b〉 ∈ f} the pre-image of B ′ under f.
c○: Michael Kohlhase 57
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