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

38 CHAPTER 3. ELEMENTARY DISCRETE MATH
#({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)
©: Michael Kohlhase 58
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 93 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 94 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 95 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,
Im(f) := f(A) the image of f, and
f −1 (B ′ ) := {a ∈ A | ∃b ∈ B ′ .〈a, b〉 ∈ f} the pre-image of B ′ under f
©: Michael Kohlhase 59