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By John Casey

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Additional resources for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples.

Example text

Geordnete Liste: 1. Zusammenfassung: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. , 7. , 8. , 7. , 9. , 8. , 9. , 7. , 10. , 8. , 10. , 9. , 10. , 11. , 11. , 11. , 11. xyz u xyzu xyzu xy zu xyzu x yzu xy z u xyz u x yzu x y zu xyzu xz u yz u xy u yzu xyu xzu xy z y zu xyz x zu x yz x yu yzu xzu xyu xyz 2. geordnete Liste: 2. Zusammenfassung: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. , 4. , 4. , 4. , 8. , 8. , 8. , 12. , 12. , 12. , 16. , 16. , 16. xy z xyz x yz xyz xy u xyu xyu xyu yz u yzu y zu yzu xz u xzu x zu xzu 53 yz xz xy yu xu xy zu yu yz zu xu xz Da weitere Zusammenfassungen nicht mehr m¨ oglich sind, erhalten wir r(x, y, z, u) = x y ∨ x z ∨ y z ∨ x u ∨ y u ∨ z u.

Sei A := {1, 2, 3}. Dann ist die Relation R1 := {1, 2), (1, 3), (2, 3)} nicht reflexiv, aber antisymmetrisch und transitiv; die Relation R2 := {1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} nicht antisymmetrisch, aber reflexiv und transitiv; und die Relation R3 := {1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} nicht transitiv, aber reflexiv und antisymmetrisch. 45 (Hintereinanderausf¨ uhren von Abbildungen) Seien A := {x ∈ R | 0 ≤ x ≤ 1} und f1 : A −→ A, x → sin x, f2 : A −→ A, x → x2 , √ f3 : A −→ A, x → x. Durch welche Zuordnungsvorschriften sind dann fi ✷fj f¨ ur {i, j} ⊂ {1, 2, 3} und i = j bestimmt?

8. , 8. , 8. , 12. , 12. , 12. , 16. , 16. , 16. xy z xyz x yz xyz xy u xyu xyu xyu yz u yzu y zu yzu xz u xzu x zu xzu 53 yz xz xy yu xu xy zu yu yz zu xu xz Da weitere Zusammenfassungen nicht mehr m¨ oglich sind, erhalten wir r(x, y, z, u) = x y ∨ x z ∨ y z ∨ x u ∨ y u ∨ z u. Man pr¨ uft leicht nach, daß in den verk¨ urzten Darstellungen f¨ ur g und r keine Konjunktionen weggelassen werden k¨ onnen. , xn ). , xn ) = a0 + a1 · x1 + a2 · x2 + ... , xn ∈ {0, 1} gilt. , bn ))) gilt. (a) (b) (c) (d) Wie viele selbstduale ein- oder zweistellige Boolesche Funktionen gibt es?

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