By Guerino Mazzola, Gérard Milmeister, Jody Weissmann
Comprises all of the arithmetic that machine scientists want to know in a single position.
Read or Download Comprehensive Mathematics for Computer Scientists 1: Sets and Numbers, Graphs and Algebra, Logic and Machines, Linear Geometry (v. 1) PDF
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Additional resources for Comprehensive Mathematics for Computer Scientists 1: Sets and Numbers, Graphs and Algebra, Logic and Machines, Linear Geometry (v. 1)
Let x ∈ a be any element different from x0 . Since d is alternative, either x ∈ x0 or x0 ∈ x. But x ∈ x0 contradicts x0 ∩ a = ∅, and we are done. Proposition 31 If d is ordinal, then x ∈ d implies that x is ordinal. Proof Let x ∈ d. Then by transitivity of d, x ⊂ d. Hence x is alternative and founded. Let b ∈ x. Then a ∈ b implies a ∈ x. In fact, x ⊂ d, therefore b ∈ d and a ∈ b ∈ d. Since d is transitive, a ∈ d. So a, x ∈ d. Hence either a = x, or a ∈ x, or x ∈ a. But by lemma 29, applied to the element chain a ∈ b ∈ x, we must have a ∈ x.
The functions f : A → B and g : B → C are both bijections. The composition of g and f is a bijection h = g ◦ f : A → C. The purpose of this example is to show that equipollence is a feature independent of the shape, or “structure” of the set. It only tells us that each element from the first set can be matched with an element from the second set, and vice-versa. Proposition 18 For all sets a, b and c, we have: (i) (Reflexivity) a is equipollent to a. (ii) (Symmetry) If a is equipollent to b, then b is equipollent to a.
9 in a graphical way. An element (x, y) ∈ R is represented by a point at the intersection of the vertical line through x and the horizontal line through y. For the reflexive, symmetric and transitive closure, added elements are shown in gray. e e e e d d d d c c c c b b b b a a a b c (a) d e a a b c (b) d e a a b c (c) d e a b c d e (d) Fig. 9. A relation R (a), its reflexive closure Rr (b), the reflexive, symmetric closure (Rr )s (c), and the reflexive, symmetric, transitive closure Re = ((Rr )s )t (d).
Comprehensive Mathematics for Computer Scientists 1: Sets and Numbers, Graphs and Algebra, Logic and Machines, Linear Geometry (v. 1) by Guerino Mazzola, Gérard Milmeister, Jody Weissmann