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

Example 75 The simplest function we always try everything on is the identity function:
λn ∈ N.n = {〈n, n〉 | n ∈ N} = IdN
= {〈0, 0〉, 〈1, 1〉, 〈2, 2〉, 〈3, 3〉, . . .}
Example 76 We can also to more complex expressions, here we take the square function
λx ∈ N.x 2 = {〈x, x 2 〉 | x ∈ N}
= {〈0, 0〉, 〈1, 1〉, 〈2, 4〉, 〈3, 9〉, . . .}
Example 77 λ-notation also works for more complicated domains. In this case we have
tuples as arguments.
λ〈x, y〉 ∈ N 2 .x + y = {〈〈x, y〉, x + y〉 | x ∈ N ∧ y ∈ N}
= {〈〈0, 0〉, 0〉, 〈〈0, 1〉, 1〉, 〈〈1, 0〉, 1〉,
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〈〈1, 1〉, 2〉, 〈〈0, 2〉, 2〉, 〈〈2, 0〉, 2〉, . . .}
4 EdNote:4
The three properties we define next give us information about whether we can invert functions.
Properties of functions, and their converses
Definition 78 A function f : S → T is called
injective iff ∀x, y ∈ S.f(x) = f(y) ⇒ x = y.
surjective iff ∀y ∈ T.∃x ∈ S.f(x) = y.
bijective iff f is injective and surjective.
Note: If f is injective, then the converse relation f −1 is a partial function.
Note: If f is surjective, then the converse f −1 is a total relation.
Definition 79 If f is bijective, call the converse relation f −1 the inverse function.
Note: if f is bijective, then the converse relation f −1 is a total function.
Example 80 The function ν : N1 → N with ν(o) = 0 and ν(s(n)) = ν(n) + 1 is a bijection
between the unary natural numbers and the natural numbers from highschool.
Note: Sets that can be related by a bijection are often considered equivalent, and sometimes
confused. We will do so with N1 and N in the future
Cardinality of Sets
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Now, we can make the notion of the size of a set formal, since we can associate members of
sets by bijective functions.
Definition 81 We say that a set A is finite and has cardinality #(A) ∈ N, iff there is a
bijective function f : A → {n ∈ N | n < #(A)}.
4 EdNote: define Idon and Bool somewhere else and import it here
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