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Transcription:

(7A)

Coyright (c) 215 218 Young W. Lim. Permission is granted to coy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version ublished by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A coy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was roduced by using LibreOffice and Octave.

Argument ( ) ( r) q r ( ) ( r) q r 3

Truth Table q r T T T T T T F T T F T T T F F T F T T T F T F T F F T F F F F F q r r T T T F T T T F F F T F T F T T F F F F F T T T T F T F T T F F T T T F F F T T q r ( ) ( r) q r T T T T T T T F F T T F T T T T F F F F F T T T T F T F T T F F T F T F F F F F ( ) ( r) q r 4

Interretation of this truth table case case case case case case case case A B q r r ( ) ( r) q r T T T T T T T T T F T F F T T F T T T T T T F F T F F F F T T T T T T F T F T T T T F F T F T F T F F F F T F F A B T T T T T T T T Whenever and r are true, q r is true ( ) ( r) q r 5

Venn diagram for case case case case case case case case q r T T T T T T F T T F T T T F F T F T T T F T F T F F T F F F F F 6

Venn diagram for r r case case case case case case case case q r T T T F T T T F F F T F T F T T F F F F F T T T T F T F T T F F T T T F F F T T 7

When ( ) ( r) is true r When and r is true is true ( ) ( r) cases +++ 8

When ( ) ( r) is true, q r is also true r q r cases +++ ( ) ( r) q r cases +++++ 9

Argument Case 1: is false Case 2: is true r q r F q T r q T q F r r when is false, q must be true. when is true, r must be true. Therefore regardless of truth value of, If both remises hold, then the conclusion q r is true htt://en.wikiedia.org/wiki/derivative 1

Examles r q r r r q r q q r r r r r 11

in Prolog Conjunctive Norm Form (CNF) is assumed ( ) ( ) ( ) the variables A, B, C, D, and E are in conjunctive normal form: A ( B C ) ( A B ) ( B C D ) ( D E ) A B A B The following formulas are not in conjunctive normal form: ( B C ) ( A B ) C A ( B ( D E ) ) htts://en.wikiedia.org/wiki/conjunctive_normal_form 12

General Rules of Inference (1) q q ( ( q)) q Discrete Mathematics and Its Alications, Rosen Modus onens Modus: manner, way, mode Ponens: to affirm q q ( q ( q)) Modus tollens Modus: manner, way, mode tollens: to remove q q r r (( q) (q r)) r Hyothetical syllogism Syllogism: inference, conclusion q (( ) ( )) q Disjunctive syllogism 13

General Rules of Inference (2) Discrete Mathematics and Its Alications, Rosen ( ) Addition q ( q) Simlification q q ( q) q Conjunction q r (( ) ( r)) q r Breaking into arts, a loosening 14

Examle A r q r q r q r r r ( ) ( r) ( ) ( q r) ( ) ( r) ( ) ( q r) (q) ( ) ( r) ( ) ( q r) (q) (r) ( ) ( r) ( ) ( q r) (q) (r) ( ) James Asnes, Notes on Discrete Mathematics, CS 22: Fall 213 15

Examle B (1) q r q r q r q r q q q r r q r q q q r = T r = T q r q r q r q r q r Discrete Mathematics, Johnsonbough 16

Examle B (2) q r q r q q r q q r q r q q r r Discrete Mathematics, Johnsonbough 17

Truth Table q r r T T T F T T T F F T T F T F T T F F F F F T T T T F T F T T F F T T T F F F T T q r q q T T T F T T T F F T T F T T T T F F T T F T T F F F T F F F F F T T T F F F T T q r q r T T T T T T F T T F T T T F F F F T T T F T F T F F T T F F F F q r r q q r H 1 H 2 H 3 r T T T T T T T T T T F T T T T T T F T T T T T T T F F F T F F T F T T T F T F T F T F T F T F F F F T T T T T T F F F T T F F F H 1 = r H 2 = q H 3 = q r H 1 H 2 H 3 H 3 H 1 H 2 H 3 H 2 H 1 H 2 H 3 H 1 H 1 H 2 H 3 ( r) 18

Truth Table and K-Ma q r H 1 H 2 H 3 T T T T T T F T T F T T T F F F F T T F F T F F F F T T F F F F q r H 1 H 2 H 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q r H 1 H 2 H 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q r 1 1 1 1 1 19

K-Ma and Logic Minimization q r 1 1 1 1 1 1 q r q r q r q r 1 q r q q r + q r = ( + ) q r = q r q r + q r = q(r + r) = q 2

K-Ma : Verification 1 q r q H 1 H 2 H 3 q r + q q r q q r q q r+ q 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q r H 1 H 2 H 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 21

Adding two don t care conditions q r q r 1 X X 1 X X 1 1 1 1 1 1 1 1 1 r 1 q q r 22

Adding two don t care conditions q r q r 1 1 X 1 1 1 1 1 X 1 1 1 1 r 1 r 23

K-Ma : Verification 1 r 1 H 1 H 2 H 3 r q r r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 q r H 1 H 2 H 3 r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 24

Examle C q q q q q q ( ) q = ( q) (q q) = ( q) Simlification Rule Discrete Mathematics, Johnsonbough 25

References [1] htt://en.wikiedia.org/ [2]

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