Logic

Grundlagen der Mathematik [Lecture notes] by Thomas Markwig Keilen

By Thomas Markwig Keilen

Show description

Read Online or Download Grundlagen der Mathematik [Lecture notes] PDF

Best logic books

Technologically Enhanced Natural Radiation

This publication on TENR discusses the fundamental Physics and Chemistry ideas of natural radiation. the present wisdom of the organic results of average radiation is summarized. a wide selection of themes, from cosmic radiation to atmospheric, terrestrial and aquatic radiation is addressed, together with radon, thoron, and depleted uranium.

Computational Logic in Multi-Agent Systems: 13th International Workshop, CLIMA XIII, Montpellier, France, August 27-28, 2012. Proceedings

This ebook constitutes the lawsuits of the thirteenth overseas Workshop on Computational common sense in Multi-Agent structures, CLIMA XIII, held in Montpellier, France, in August 2012. The eleven common papers have been rigorously reviewed and chosen from 27 submissions and provided with 3 invited papers. the aim of the CLIMA workshops is to supply a discussion board for discussing suggestions, in accordance with computational good judgment, for representing, programming and reasoning approximately brokers and multi-agent structures in a proper means.

Computational Logic in Multi-Agent Systems: 8th International Workshop, CLIMA VIII, Porto, Portugal, September 10-11, 2007. Revised Selected and Invited Papers

This booklet constitutes the completely refereed post-conference complaints of the eighth foreign Workshop on Computational good judgment for Multi-Agent structures, CLIMA VIII, held in Porto, Portugal, in September 2007 - co-located with ICLP 2008, the foreign convention on common sense Programming. The 14 revised complete technical papers and 1 method description paper awarded including 1 invited paper have been rigorously chosen from 33 submissions and went via a minimum of rounds of reviewing and development.

Logic and the Nature of God

The ebook '. .. might be guaranteed of the eye of the various on either side of the Atlantic who're excited about this topic. ' John Hick

Additional resources for Grundlagen der Mathematik [Lecture notes]

Example text

Wir f¨ uhren das f¨ ur die Addition der rationalen Zahlen vor. Sind (p ′ , q ′ ) ∈ (p, q) und (r ′ , s ′ ) ∈ (r, s) andere Repr¨asentanten, dann gilt p ′ q = q ′ p und r ′ s = s ′ r. Es ist zu zeigen, daß (p ′ s ′ + q ′ r ′ , q ′ s ′ ) ∈ (ps + qr, qs) gilt. Ausmultiplizieren liefert (p ′ s ′ + q ′ r ′ )(qs) = p ′ qs ′ s + q ′ qr ′ s = q ′ ps ′ s + q ′ qs ′ r = (ps + qr)(q ′ s ′ ), was zu zeigen war. 13 Wir definieren f¨ ur zwei Punkte (x, y), (x ′ , y ′ ) ∈ R2 (x, y) ∼ (x ′ , y ′ ) :⇐⇒ |x| + |y| = |x ′ | + |y ′ |.

N + 1 − k)! · k! n+1 . 15 (Binomischer Lehrsatz) Es sei K ein K¨orper, x, y ∈ K und n ∈ N, so gilt n n · xk · yn−k . k n (x + y) = k=0 Beweis: Wir f¨ uhren den Beweis durch Induktion nach n. Induktionsanfang: n = 0: Nach Definition gilt 0 0 · xk · y0−k . k 0 (x + y) = 1 = 1 · 1 · 1 = k=0 Induktionsschluß: n → n + 1: Es gilt (x + y)n+1 =(x + y)n · (x + y) = (x + y)n · x + (x + y)n · y n Ind. 14 n+1 =x + k=1 n+1 = k=0 n k=0 n · xk · yn+1−k k n · xk+1 · yn−k + k n k=1 n · xk · yn+1−k + k−1 n · xk · yn+1−k + yn+1 k n k=1 n · xk · yn+1−k + yn+1 k n+1 · xk · yn+1−k + yn+1 k n+1 · xk · yn+1−k k Die Aussage folgt damit aus dem Prinzip der vollst¨andigen Induktion.

Seien dazu x = (p, q), x = (p , q ), x = (p , q ) ∈ M gegeben:1 R1: F¨ ur die Reflexivit¨at m¨ ussen wir x ∼ x zeigen. Nun gilt aber pq = pq, woraus x = (p, q) ∼ (p, q) = x folgt. R2: F¨ ur die Symmetrie nehmen wir an, daß x ∼ x ′ gilt und m¨ ussen x ′ ∼ x folgern. Wegen x ∼ x ′ gilt aber nach Definition pq ′ = p ′ q, und folglich auch p ′ q = pq ′ . Letzteres bedeutet aber, daß x ′ = (p ′ , q ′ ) ∼ (p, q) = x. R3: F¨ ur die Transitivit¨at nehmen wir schließlich an, daß x ∼ x ′ und x ′ ∼ x ′′ gilt, und m¨ ussen daraus schließen, daß x ∼ x ′′ .

Download PDF sample

Rated 4.67 of 5 – based on 7 votes