 By Gérard Milmeister, Guerino Mazzola, Jody Weissmann

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Additional info for Comprehensive Mathematics for Computer Scientists 1: Sets and Numbers, Graphs and Algebra, Logic and Machines, Linear Geometry (Universitext)

Sample text

Exercise 12 Show that the induced relation R|a is a partial ordering, a linear ordering, a well-ordering, if R is so. Exercise 13 Given a relation R on a and a bijection f : a → b, then we consider the image Rf of the induced bijection (f × f )|R in b × b. This new relation is called “structural transport” of R. , it is a partial ordering, a linear ordering, a wellordering, iff R is so. The strongest statement about relations on sets is this theorem (due to Ernst Zermelo): Proposition 27 (Zermelo) There is a well-ordering on every set.

2. Alternativity: In an alternative set a, if x ≠ y are two elements of a, then either x ∈ y or y ∈ x. Definition 23 A set a is called founded if for each non-empty b ⊂ a, there is x ∈ b with x ∩ b = ∅. Example 16 The sets 0 and 1 are founded. What does it mean for a set not to be founded. Consider the negation of foundedness: If a set a is not founded, then there is “bad” non-empty subset b ⊂ a, such that for all x ∈ b, we have x ∩ b ≠ ∅, in other words: every element of b has an element, which already is in b.

V Proof We have the set v = a ∪ b. Let P = 2(2 ) be the powerset of the powerset of v, which also exists. Then an ordered pair (x, y) = {{x}, {x, y}}, with x ∈ a and y ∈ b is evidently an element of P . ” Sorite 11 Let a, b, c, d be sets. Then: (i) a × b = ∅ iff a = ∅ or b = ∅, (ii) if a × b ≠ ∅, then a × b = c × d iff a = c and b = d. Proof The first claim is evident. As to the second, if a × b ≠ ∅, then we have a ∪ b = ( (a × b)), as is clear from the definition of ordered pairs. Therefore we have the subset a = {x | x ∈ a ∪ b, there is z ∈ a × b with z = (x, y)}.