By G Mazzola; Gérard Milmeister; Jody Weissmann

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Definition 10 A function is a triple (a, f , b) such that f is a functional graph, where a = pr 1 (f ) and pr 2 (f ) ⊂ b. The set a is called the domain of the function, the set b is called its codomain, and the set pr 2 (f ) is called the function’s image and denoted by Im(f ). One usually denotes a function by a more graphical sign f : a → b. For x ∈ a, the unique y ∈ b such that (x, y) ∈ f is denoted by f (x) and is called the value of the function at the argument x. Often, if the domain and codomain are clear, one identifies the function with the graph sign f , but this is not the valid definition.

If we had c ∈ a, then we cannot have a ∩ {a, b, c} = ∅. By the hypothesis of the lemma, b ∩ {a, b, c} contains a, and c ∩ {a, b, c} contains b. Therefore the set {a, b, c} contradicts the foundedness of d and so c ∈ a. Here is the fundamental concept for creating numbers: Definition 24 A set is called ordinal if it is transitive, alternative, and founded. The following series of results is destined to control the basic properties of ordinals. 48 Ordinal and Natural Numbers Lemma 30 Let d be an ordinal.

For each relation R on a, the inverse relation R −1 = {(y, x) | (x, y) ∈ R} (the inverse graph) is a relation on a. If R and S are two relations on a, then the composed graph R ◦ S defines the composed relation on a. In particular, we have the second power R 2 = R ◦ R of a relation R. Notation 4 Often, relation symbols are not letters, but special symbols such as <, ≤, ≺, . . Their usage is completely dependent on context and has no universal meaning. Given relations < and ≤, the corresponding inverse relations are denotated by > and ≥, respectively.