1.(p  q) 2. (r  s) 3. p v r  q v s 4. (p  q)  (r  s) conj. 1,2 5. q v sC.D. 3,4.

Slides:

Advertisements

Similar presentations

Propositional Equivalences

Advertisements

Propositional Equivalences

1 Section 6.3 Formal Reasoning A formal proof (or derivation) is a sequence of wffs, where each wff is either a premise or the result of applying a proof.

CSE 20 Lecture 9 Boolean Algebra: Theorems and Proofs CK Cheng April 26, 2011 Lecture notes 1.

1 Introduction to Abstract Mathematics Valid AND Invalid Arguments 2.3 Instructor: Hayk Melikya

AB 11 22 33 44 55 66 77 88 99 10  20  19  18  17  16  15  14  13  12  11  21  22  23  24  25  26  27  28.

1/03/09 De 89 à 98. 1/03/09 De 89 à 98 1/03/09 De 89 à 98.

Fig. 16-CO, p Fig. 16-1, p. 450 Fig. 16-2, p. 450.

Laws of , , and  Excluded middle law Contradiction law P   P  T P   P  F NameLaw Identity laws P  F  P P  T  P Domination laws P  T  T P.

1 CA 208 Logic Ex4 Commutativity (P  Q)  Associativity P  (Q  R)  (P  Q)  R Distributivity P  (Q  R)  Idempotency.

1 Inference Rules and Proofs Z: Inference Rules and Proofs.

1 Inference Rules and Proofs Z: Inference Rules and Proofs.

Proofs Using Logical Equivalences Rosen 1.2 List of Logical Equivalences p  T  p; p  F  pIdentity Laws p  T  T; p  F  FDomination Laws p  p.

Chapter 24: Some Logical Equivalences. Logically equivalent statement forms (p. 245) Two statements are logically equivalent if they are true or false.

   a + 8a. Objective: To add and subtract real numbers using rules.

Propositional Proofs 2.

Deduction, Proofs, and Inference Rules. Let’s Review What we Know Take a look at your handout and see if you have any questions You should know how to.

7.4 Rules of Replacement II Trans, Impl, Equiv, Exp, Taut.

Philosophy 200 substitution rules. Substitution Sometimes, when you translate a statement from English to SL, you translate it in a form that is less.

DeMorgan’s Rule DM ~(p v q) :: (~p. ~q)“neither…nor…” is the same as “not the one and not the other” ~( p. q) :: (~p v ~q) “not both…” is the same as “either.

Mathematical Structures for Computer Science Chapter 1.2 Copyright © 2006 W.H. Freeman & Co.MSCS SlidesFormal Logic.

$100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300.

Thinking Mathematically Logic 3.6 Negations of Conditional Statements and De Morgan’s Laws.

Transposition (p > q) : : ( ~ q > ~ p) Negate both statements when switching order of antecedent and consequent If the car starts, there’s gas in the tank.

Chapter Four Proofs. 1. Argument Forms An argument form is a group of sentence forms such that all of its substitution instances are arguments.

Thinking Mathematically Equivalent Statements, Conditional Statements, and De Morgan’s Laws.

1 Introduction to Abstract Mathematics Expressions (Propositional formulas or forms) Instructor: Hayk Melikya

assumption procedures

Formal Logic Mathematical Structures for Computer Science Chapter 1 Copyright © 2006 W.H. Freeman & Co.MSCS SlidesFormal Logic.

UMBC CMSC 203, Section Fall CMSC 203, Section 0401 Discrete Structures Fall 2004 Matt Gaston

BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION

CHAPTER 1 INTRODUCTION TO DIGITAL LOGIC

Symbolic Logic ⊃ ≡ · v ~ ∴. What is a logical argument? Logic is the science of reasoning, proof, thinking, or inference. Logic allows us to analyze a.

CHAPTER 1 INTRODUCTION TO DIGITAL LOGIC. De Morgan’s Theorem De Morgan’s Theorem.

DE MORGAN’S THEOREM. De Morgan’s Theorem De Morgan’s Theorem.

Today’s Topics Argument forms and rules (review)

1 Propositional Proofs 1. Problem 2 Deduction In deduction, the conclusion is true whenever the premises are true.  Premise: p Conclusion: (p ∨ q) 

Law of logic Lecture 4.

。 33 投资环境 3 开阔视野提升竞争力。 3 嘉峪关市概况。 3 。 3 嘉峪关是一座新兴的工业旅游城市，因关得名，因企设市，是长城文化与丝路文化交汇点，是全国唯一一座以长城关隘命名的城市。嘉峪关关城位于祁连山、黑山之间。 1965 年建市，下辖雄关区、镜铁区、长城区，全市总面积 2935.

L = # of lines n = # of different simple propositions L = 2 n EXAMPLE: consider the statement, (A ⋅ B) ⊃ C A, B, C are three simple statements 2 3 L =

Lecture 1-2: Propositional Logic. Propositional Logic Deriving a logical conclusion by combining many propositions and using formal logic: hence, determining.

March 23 rd. Four Additional Rules of Inference  Constructive Dilemma (CD): (p  q) (r  s) p v r q v s.

7.1 De Morgan's Laws Bond Chapter 7.1 Part 2.

2. The Logic of Compound Statements Summary

De Morgan’s Theorem,.

More on Logical Equivalence

Methods of proof Section 1.6 & 1.7 Wednesday, June 20, 2018

Natural Deduction: Using simple valid argument forms –as demonstrated by truth-tables—as rules of inference. A rule of inference is a rule stating that.

A v B ~ A B > C ~C B > (C . P) B / ~ B mt 1,2 / B ds 1,2

تصنيف التفاعلات الكيميائية

Double Negation p :: ~ ~p 1.A > ~ (B . C) 2. B . C ~ ~ ( H v K) ~ H

Propositional Equivalences

Information Technology Department

7.1 Rules of Implication I Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic.

Deductive Reasoning: Propositional Logic

Natural Deduction 1 Hurley, Logic 7.4.

CSE 311 Foundations of Computing I

8.2 Using the Rule of Inference

More on Logical Equivalence

Malini Sen WESTERN LOGIC Presented by Assistant Professor

1. (H . S) > [ H > (C > W)]

Discrete Mathematics Lecture # 3.

Boolean Algebra How gates get picked.

Natural Deduction Hurley, Logic 7.2.

Natural Deduction Hurley, Logic 7.3.

Transposition (p > q) : : ( ~ q > ~ p) Negate both statements

Chapter 8 Natural Deduction

Summary of the Rules A>B -A B -(A>B) A -B --A A AvB A B

If v.s. Only If v.s. If and Only If (Iff)

Presentation transcript:

1.(p  q) 2. (r  s) 3. p v r  q v s 4. (p  q)  (r  s) conj. 1,2 5. q v sC.D. 3,4

1. p  q 2. q  s 3. s  m 4. p  m 5.qm.p. 1,4 6. s m.p. 2,5 7. mm.p. 3,6

1. p  q 2. q  r  p  ( p  r) 3. p  r H.S. 1,2 4. p  ( p  r) Abs. 3

1. p  q 2. q  m 3. m  k 4. k  s 5. ~s v n 6. ~n  ~p v (n  k) 7. ~s d.s. 5, 6 8. ~k m.t. 4, 7 9. ~m m.t. 3, ~qm.t. 2, ~pm.t. 1, ~p v (n  k) add 11

1. j  k 2. k v m 3. (m  ~j)  (~q  k) 4. ~k  ~q 5. m d.s. 2, 4 6. ~j m.t. 1, 4 7. m  ~j conj. 5, 6 8. ~q  km.p. 3, 7 9. ~qSimp. 8

1. a  b 2. b  c 3. k  m 4. a v k 5. ~c  m 6. a  c h.s. 1, 2 8. c v mc.d. 7,4 9. m d.s. 8, 5 7. (a  c) (k  m ) Conj. 6, 3

1. a  b 2. a v (c  d) 3. ~b  ~e  c 4. ~b Simp ~a m.t. 4, 1 6. c  d d.s. 2,5 7. cSimp. 6

1. w  x 2. (w  x)  y 3. (w  y)  z  w  (w  z) 5. w  (w  x ) abs w  y h.s 5,2 7. w  (w  y) abs w  z h.s. 7,3 9. w  (w  z) Abs. 8

1. (n v o)  p 2. (p v q)  r 3. q v n 4. ~q  r 5. n d.s. 3,4 6. n v o Add p m.p. 6, 1 8. p v qAdd rm.p. 2,8

Now we add the Equivalence Rules

1. ~p v q 2. ~(q v m)  ~p 3. p  q Material Implication 1 4. ~q  ~m De Morgen’s 2 5. ~q Simp ~pM.T. 3, 5

1. (p  q)  r 2. ~p v q 3. (z  p)  (r v s) v (z  p) 4. p  q Material Implication 2 5. rm.p. 4, 5 6. r v sAdd (r v s) v (z  p) add. 6

1. (h  i)  (j  k) 2. (i v k)  L 3. ~ L  ~(h v j) 4. ~(i v k)m.t. 2, 3 5. (h  i) Simp ~i  ~k DeMorgan’s 4 7. ~i Simp ~h m.t. 7, 5 9. ~k  ~i Commutation ~k Simp (j  k)  (h  i) Commutation (j  k) Simp ~j m.t. 10, ~j  ~h Conj. 13, ~h  ~j Comm ~(h v j) De Morgan’s 15

1. (k v p) v x 2. k  ~o 3. (p v x)  ~l  ~(o  l) 4. k v (p v x)Association 1 5. ~o v ~l C.D. 4,2,3 6. ~(o  l) De Morgan’s 5

1. ~(~p  q) 2. ~q  r 3. p  ~s  r v ~s 4. ~(~p  ~~q) Double negation 1 5. ~~(p v ~q) DeMorgan’s 4 6. p v ~q d.n ~s v r C.D. 2,3,6 8. r v ~s Commutation 7

1. ~b 2. ~(c  b)  c 3. ~ g  ~c  g 4. ~b v ~cAdd ~c v ~b Comm ~(c  b) DeMorgan’s 5 7. c m.p. 2, 6 8. c  g Contraposition (Transposition) 3 9. g m.p. 7, 8

1. p  q 2. ~(p v m)  ~(z  k) 3. ~m 4. ~q  ~ z v ~ k

1. b  e 2. ~m v g 3. (b  c)  d 4. (d  c)  m  e  g