<< /S /GoTo /D (section.2) >> >> endobj /Rect [132.772 393.676 240.397 404.525] /Border[0 0 0]/H/I/C[1 0 0] 25 0 obj AP Bio: EVO (BI), EVO‑1 (EU), EVO‑1.F (LO), EVO‑1.F.1 (EK), EVO‑1.G (LO), EVO‑1.G.1 (EK) Learn. Natural Deduction. The specific system used here is the one found in forall x: Calgary Remix. >> endobj 3 Whom is it addressed to; 1. 4,6 (Use the following tabs if you need he rbering any of the natural deduction rules you have leamed so far.) + The precision of formal languages avoid the ambiguities of natural lan-guages.] We need a deductive system, which will allow us to construct proofs of tautologies in a step-by-step fashion. /D [114 0 R /XYZ 132.768 705.06 null] It is used particularly to present the syntax of formal logic and type theory. endobj Logical terminology is generally difficult to understand in a short period of time, and the use of quizzes helps to provide some measure of conceptual understanding prior to tests. /Border[0 0 0]/H/I/C[1 0 0] /Parent 181 0 R 33 0 obj Natural deduction proof editor and checker. /Subtype /Link CONSTRUCTING CORRECT DERIVATIONS Knowing the rules for constructing derivations is one thing. (C.) ( PC). I Best way to study will be through practice questions in forallx . >> endobj endobj /Subtype /Link 141 0 obj << 156 0 obj << endobj /Rect [461.539 134.592 478.476 143.005] >> endobj August 2004 (reviewed at May 2005) Contents; 1 Before starting.... 1. 7.4 Aplia Assignment X 1. /Subtype /Link >> endobj %PDF-1.5 In this respect, the two systems are very similar. stream >> endobj (CW) ( PC). Tweet. endobj Q^Y )W 4. >> endobj 162 0 obj << I this clip we consider five questions where the solution to each question demonstrates a basic type of argument. 4. /Subtype /Link /Rect [147.716 216.341 222.159 227.19] /Type /Annot So I'm new to logic and taking an introductory logic course, and I'm really having trouble with these 2 questions: Using the system of Natural Deduction in the textbook, provide a derivation to establish that the following sentence is a Logical Truth: A ⊃ (B ⊃ A) I NP 3. Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system).Such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise Principia Mathematica.Spurred on by a series of seminars in Poland in 1926 by Łukasiewicz … /Subtype /Link (WP) 1 5. 2. endobj Provide the definition of maximally consistent set of formulas and show that if … 2,4 7. 3. Describe each step and which labeled rules have been applied. endobj /A << /S /GoTo /D (section.4) >> /Rect [132.772 473.378 238.771 484.226] Ubuntu 20.04 LTS. /Type /Annot /Subtype /Link (A→A) → (B→B) 2. << /S /GoTo /D (subsection.5.3) >> >> endobj /Subtype /Link 101 0 obj Here is how the proof checker associated with the forallx text presents this: Because one is trying to derive a conditional, assume the antecedent of the conditional which is what is done on line 2. /Subtype /Link /Border[0 0 0]/H/I/C[1 0 0] 2 Used symbols; 2. /Contents 172 0 R /Rect [470.755 453.397 478.476 461.81] endobj >> endobj 4 Notation. (Core) | 150 0 obj << 1,2 NATURAL DEDUCTION RULES AND PR THODS 2 Modus Tollens (MT) Modus Ponens (MP) Simplification (Simp) Conjunction (Conj) Double Negation (DN) Pure Hypothetical Syllogism (HS) Disjunctive Syllogism (DS) Constructive Dilemma (CD) Addition (Add) De Morgan's Rule (DM) Commutativity (Com) Associativity (Assoc) Transposition (Trans) Material Implication (Impl) Material Equivalence (Equiv) Exportation (Exp) Distribution (Dist) Tautology (Taut) Modus ponens (MP): pa р 9 Explanation: If p implies 4, and if you have p, you can obtain q. Grade It Now Save & Continue Both the premises and the conclusion may contain meta-variables representing arbitrary propositions. 1 1. 117 0 obj << (Negation) /Border[0 0 0]/H/I/C[1 0 0] >> endobj /Border[0 0 0]/H/I/C[1 0 0] /Border[0 0 0]/H/I/C[1 0 0] << /S /GoTo /D (subsection.4.4) >> (~ CP). /Subtype /Link >> /Border[0 0 0]/H/I/C[1 0 0] (~CP). 1 0 obj /Type /Annot << /S /GoTo /D (subsection.4.9) >> 40 0 obj 7. /Border[0 0 0]/H/I/C[1 0 0] /Border[0 0 0]/H/I/C[1 0 0] >> endobj /A << /S /GoTo /D (subsection.4.1) >> Screenshots. Q 5. :(:X) 6. 138 0 obj << /A << /S /GoTo /D (subsection.4.3) >> @�@��e[� Chain ruleα,β,γ is assumed as an axiom scheme, stating that sentence (α ⊃ β ⊃ γ) ⊃ (α ⊃ β) ⊃ α ⊃ γ is expected to be deducible, instantiated for any subsentences α,β 9.2.1 Solutions to Fill in the Blank Exercises; 9.3 Exercises: Two sets; 9.4 Rules of Equivalence; 9.4.1 Listing of the rules of inference and equivalence. << /S /GoTo /D (subsection.5.5) >> CV NW 4. << /S /GoTo /D (subsection.5.2) >> 41 0 obj /Type /Annot This is a great example for walking you through what we are introducing in this chapter, called Natural Deduction — deducing things in a “natural way” from what we already know, given a set of rules we know we can trust. 6. endobj Diagram notation conventions for analytical reasoning setups. /A << /S /GoTo /D (section.1) >> endobj 60 0 obj /A << /S /GoTo /D (subsection.5.8) >> Completing complex natural deduction proofs requires the ability to recognize basic argument patterns in groups of compound statements and often requires that you "reason backward" from the conclusion to identify what rules will allow you to obtain the conclusion and in what order they must be applied. 3. Natural Deduction - Practice 2 As you learn additional natural deduction rules, and as your ability to think several steps ahead to determine complex natural deduction proofs requires the ability t you "reason backward" from the conclusion to identify Developing these skills requires regular practice and re of each page in this problem set. /Rect [471.502 417.531 478.476 425.944] /Border[0 0 0]/H/I/C[1 0 0] (Additional challenges) /Border[0 0 0]/H/I/C[1 0 0] 145 0 obj << /A << /S /GoTo /D (subsection.4.3) >> >> endobj CVW 4. Just as in the truth tree system, we number the statements and include a justification for every line. /Length 2812 134 0 obj << endobj /Subtype /Link /A << /S /GoTo /D (subsection.4.7) >> /Rect [466.521 288.017 478.476 296.43] /Subtype /Link 3 Precedence of operators. Answer for question: Your name: Answers. There are no simple mechanical guidelines to tell you which rule to apply next, so constructing derivations is a matter of skill and ingenuity. << /S /GoTo /D (subsection.5.10) >> /Subtype /Link Free e-mail watchdog. /Rect [147.716 156.566 264.169 167.414] 3. /A << /S /GoTo /D (section.5) >> (C.) ( PC). /Subtype /Link /Border[0 0 0]/H/I/C[1 0 0] 80 0 obj Most of the deduction rules come in one of two flavors, introduction or elimination. >> endobj 113 0 obj Strengthen your cognitive abilities, all answers are explained. 123 0 obj << We present two core components, namely solution generation and practice problem generation, for enabling computer-aided education for this important subject domain. ( PW) (~CP) (WP) 5. /Type /Annot >> 32 0 obj /Type /Annot << /S /GoTo /D (subsection.4.3) >> 69 0 obj << /S /GoTo /D [114 0 R /Fit] >> 153 0 obj << /Type /Annot w6[�JO��x����An� ��a���';U�;͆kSW���}�D�=Y��2����F�n�j����E/�����n�=�/�h�b�'SΌY�t��ML For one, the natural deduction system also has no branching rules. 5. 168 0 obj << /Type /Annot /Rect [147.716 345.856 222.63 356.593] 13 0 obj ( PW) 2. The questions will quiz you on how your tax liability is calculated and an important aspect of the tax code. >> endobj endobj /A << /S /GoTo /D (subsection.5.1) >> /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link >> endobj Proof generator and proof checker for propositional logic in "natural deduction" style. 2 Basic concepts. /Rect [147.716 415.594 264.169 426.442] /Subtype /Link �6a��(��6���Oр��d��3�-���(�M���ɮ+�ʡ~��uE �Bz캢@�캢� �T��]ю�C[���3������o%캢{x1���uE��w�躢ML��|��㮨��� .1B�$D�������_��v< endobj The specific system used here is the one found in forall x: Calgary Remix. 1. endobj >> endobj EXERCISES BOOKLET forthe LogicManual óþÕŸ/óþÕÉ erearenochangestotheexercises fromlastyear’sedition VolkerHalbach Oxford ìrdAugustóþÕŸ endobj Being able to apply the rules successfully is another. /Rect [147.716 180.476 211.643 191.213] endobj 9 0 obj /A << /S /GoTo /D (subsection.4.5) >> /A << /S /GoTo /D (subsection.5.6) >> /Type /Annot 177 0 obj << Developing these skills requires regular practice and repetition completing increasingly complex proofs. Weknowtheanswer. 3 Functioning; 3. /Border[0 0 0]/H/I/C[1 0 0] 1. >> endobj 5. 128 0 obj << We stuffed all of this into the LMS. Free Python 3.9. /Rect [132.772 495.295 227.233 506.144] endobj /Rect [147.716 242.189 193.129 250.989] 143 0 obj << 105 0 obj /Border[0 0 0]/H/I/C[1 0 0] 2 Used symbols; 2. /Rect [466.521 347.793 478.476 356.206] >> endobj 9.1.1 Solutions to Pattern Recognition exercise. /Resources 171 0 R /Type /Annot >> endobj ( PC) ( PW) rup 3. endobj >> endobj /A << /S /GoTo /D (subsection.5.10) >> & /Border[0 0 0]/H/I/C[1 0 0] (Summary of rules) I recorded hours of video lectures covering the content. << /S /GoTo /D (subsection.4.8) >> This is a demo of a proof checker for Fitch-style natural deduction systems found in many popular introductory logic textbooks. /Rect [465.026 395.614 478.476 404.026] /Rect [470.755 497.233 478.476 505.645] This is a demo of a proof checker for Fitch-style natural deduction systems found in many popular introductory logic textbooks. 164 0 obj << /D [114 0 R /XYZ 133.768 667.198 null] >> endobj Daniel Clemente Laboreo. 163 0 obj << endobj endobj << /S /GoTo /D (subsection.3.2) >> 9.1.2 Learning to draw inferences; 9.2 Fill in the Blank Exercises. /Rect [466.521 359.748 478.476 368.161] /Rect [466.521 206.323 478.476 214.736] >> endobj endobj /Subtype /Link /Subtype /Link >> endobj 142 0 obj << /Rect [466.52 383.658 478.476 392.071] /Subtype /Link /Subtype /Link (C) (WP) PVP 6. endobj /A << /S /GoTo /D (subsection.5.8) >> 56 0 obj /Type /Annot /Type /Annot /Type /Annot Questions on Natural deduction proof: 1. /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link /A << /S /GoTo /D (subsection.3.1) >> ( PW) 2. /A << /S /GoTo /D (section.3) >> 4 License. 154 0 obj << >> endobj /Border[0 0 0]/H/I/C[1 0 0] 6. /Border[0 0 0]/H/I/C[1 0 0] endobj /Rect [147.716 335.838 230.6 344.749] Just as in the truth tree system, we number the statements and include a justification for every line. Daniel Clemente Laboreo. 4 The derivation rules. I am new to natural deduction and upon reading about various methods online, I came across the rule of bottom-elimination in the following example. For questions concerning natural deduction, a formal proof system studied in proof theory. 149 0 obj << /Subtype /Link 3 NATURAL DEDUCTION RULES AND PR THODS 1,4 oraz Modus Tolen Conjunction Modus Ponens (MP) Simplification (Simp) Distribution (Dist) Tautology Trut) Pure Hypothetical Syllogism (HS) Disjunctive Sylogism (DS) Constructive Dilemma (CD) Addition (Add) De Morgan's Rule (DM) Commutativity (Com) Associativity (Assoc) Transposition (Trans) Material Implication (Impl) ) Material Equivalencs (Equiv) Exportation (Exp) i Doublo Negation 1,2 3,5 Modus ponens (MP): pa P 9 Explanation: If p implies g, and if you have o, you can obtain q. Grade It Now Save & Continue neiu with /D [114 0 R /XYZ 133.768 538.079 null] ��G�8�d������CkZ,U�~J��@��'���f�h��-������萤�� �a¿�p_1�ہ���@X� /Subtype /Link NP HS DS (Use the following tabs if you need help reme any of the natural deduction rules you have learned so far.) /Border[0 0 0]/H/I/C[1 0 0] 166 0 obj << /Border[0 0 0]/H/I/C[1 0 0] Show More. /Border[0 0 0]/H/I/C[1 0 0] /Subtype /Link 16 0 obj NP /Subtype /Link endobj 2) Express the following sentence in proposition logic. << /S /GoTo /D (subsection.5.8) >> /Subtype /Link /A << /S /GoTo /D (section.5) >> PVNP MT 7. /Subtype /Link endobj /MediaBox [0 0 612 792] 151 0 obj << CVW (CwP) ( PW A-Z 4. 136 0 obj << Derive the following formulas via Natural Deduction, ¬(A⊃¬B)⊃(A∧B) Exercise 3 (Propositional logic: theory (max 3 marks). /Rect [471.502 429.487 478.476 437.899] A deductive reasoning test may be part of your assessment if you are applying for jobs within science and IT, such as technical design, engineering, and software development. /Type /Annot bĺ���^�LǺ�w�M��fY�كۛ���_�Jb�_I�DJ7E*_J�ۚ����l��'7���L�y�����h� �����$�T�ˎ#���8E\�|�����lFdq(�ǫ�w6W���wׯ�Dg��p�^�����x������C�YV#=���l�&�,��C�ZXy�����ƭzˬ��]M�;n=�9��=��4�ɜ/���`��箧x�2B�`����cbc�3�Ù�J�7�>)���Lʹ�N���#���6�O�γ�3Z�J�Ñ�����tN�8F���C�iuH$��q3�1�0t�D�06�3st? >> endobj /A << /S /GoTo /D (subsection.5.6) >> endobj endobj endobj Tweet. 135 0 obj << 114 0 obj << P 1 (Use the following tabs if you need he 1,3 mbering any of the natural deduction rules you have leamed so far.) 157 0 obj << CVW (CP) • (PW) A-Z 4. 9.1 Pattern Recognition Exercises. /Rect [466.521 218.279 478.476 226.691] /Border[0 0 0]/H/I/C[1 0 0] 160 0 obj << (Biconditional) This is the currently selected item. /A << /S /GoTo /D (subsection.5.7) >> endobj Q^Y )W 4. /Rect [132.772 451.46 237.941 462.308] endobj >> endobj /Subtype /Link © 2003-2020 Chegg Inc. All rights reserved. /Border[0 0 0]/H/I/C[1 0 0] The meta-variables are replaced consistently with the appropriate kind of proposition when an inference rule is used as part of a proof. Deductive reasoning tests are used as part of assessing candidates applying to entry and midlevel positions requiring deductive reasoning ability. endobj >> endobj (Existential quantifier) /Length 806 View desktop site, 6. ABOUT; FIND THE ANSWERS. (-CO-P). 116 0 obj << /A << /S /GoTo /D (subsection.4.4) >> /Type /Annot 119 0 obj << / -P 3. /Subtype /Link (Practice problems) 73 0 obj ( PW) w||| MP 5. Y Prove using natural deduction R^W. Answer this question. /A << /S /GoTo /D (subsection.3.2) >> 93 0 obj (Existential quantifier) 3. /A << /S /GoTo /D (subsection.5.2) >> 97 0 obj 148 0 obj << 77 0 obj << /S /GoTo /D (section.5) >> endobj >> endobj Natural deduction practice? /Type /Annot >> endobj /Subtype /Link /Type /Annot 81 0 obj 2 Why do I write this; 1. 1. endobj /Rect [466.521 170.458 478.476 178.871] /Rect [466.521 323.883 478.476 332.295] So I'm new to logic and taking an introductory logic course, and I'm really having trouble with these 2 questions: Using the system of Natural Deduction in the textbook, provide a derivation to establish that the following sentence is a Logical Truth: A ⊃ (B ⊃ A) CENGAGE MINDTAP a Search this course ? natural deduction. NATURAL DEDUCTION RULES AND PR 1,2 THODS Modus Tollen 3 Modus Ponens (MP) Simplification (Simp) Conjunction (Conj) Pure Hypothetical Syllogism (HS) Disjunctive Syllogism (DS) Constructive Dilemma (CD) Addition (Add) De Morgan's Rule (DM) Commutativity (Com) Associativity (Assoc) Transposition (Trans) Material Implication (Impl) Material Equivalence (Equiv) Exportation (Exp) Distribution (Dist) Double Negation (DN) Tautology (Taut) Modus ponens (MP): pg р 9 Explanation: If p implies q, and if you have p, you can obtain q. Grade It Now Save & Continue continue without wine /Type /Annot They diverge, however, in two important ways. 1 Formalization; 2. View natural deduction practice problem answers.pdf from PHIL 0070 at New York University. /A << /S /GoTo /D (subsection.5.5) >> /Border[0 0 0]/H/I/C[1 0 0] CENGAGE MINDTAP Q Search this course ? /Border[0 0 0]/H/I/C[1 0 0] (Implication) /Subtype /Link (Universal quantifier) Free Python 3.8. /Rect [466.521 230.234 478.476 238.647] 1 Questions & Answers Place. No. 7.4 Aplia Assignment X 1. 4 0 obj Natural deduction practice? Natural Deduction ... examples | rules | syntax | info | download | home: Last Modified : 02-Dec-2019 (B→C) → (A→A) / conclusion: (B→B) I was able to solve it using indirect proof but I want to try to prove it using the rules of inference and (Universal quantifier) /Border[0 0 0]/H/I/C[1 0 0] (CW) ( PC). 133 0 obj << /Border[0 0 0]/H/I/C[1 0 0] 76 0 obj 21 0 obj endobj /A << /S /GoTo /D (subsection.4.10) >> The key enabling technology […] /Subtype /Link /Type /Annot endobj endobj /Border[0 0 0]/H/I/C[1 0 0] %���� 122 0 obj << /Type /Annot The framework of natural deduction describes a particular class of deductive systems which is supposed to be close to “natural” deductive reasoning insofar it is based on the idea of reasoning from assumptions in contrast to proof systems that reason from ‘truths’ in the tradition of Hilbertian axiomatics. /Border[0 0 0]/H/I/C[1 0 0] /A << /S /GoTo /D (subsection.4.9) >> /Subtype /Link (Implication) /A << /S /GoTo /D (subsection.4.7) >> >> endobj (Existential quantifier) /A << /S /GoTo /D (subsection.5.3) >> 72 0 obj /A << /S /GoTo /D (subsection.4.10) >> 9.1.2 Learning to draw inferences; 9.2 Fill in the Blank Exercises. 84 0 obj ... Let's Practice Natural Deduction! 68 0 obj 1. 48 0 obj (~C-P). /Filter /FlateDecode (WP) PvP 6. /Subtype /Link 7. Answer for question: Your name: Answers. /Type /Annot Shawn designed and implemented a whole raft of multiple choice practice questions, and we worked on a range of class activities to help our class of 60 students grapple with the material. 130 0 obj << /Type /Annot endobj /A << /S /GoTo /D (subsection.3.2) >> Formalize the following sentences: 1. “xbeing prime is a sufficient condition for xbeing odd” Natural deduction, which is a method for establishing validity of propositional type arguments, helps develop important reasoning skills and is thus a key ingredient in a course on introductory logic. Natural Deduction - Practice 1 As You Learn Additional Natural Deduction Rules, And As The Proofs You Will Need To Complete Become More Complex, It Is Important That You Develop Your Ability To Think Several Steps Ahead To Determine What Intermediate Steps Will Be Necessary To Reach The Argument's Conclusion. /Border[0 0 0]/H/I/C[1 0 0] /Font << /F15 175 0 R /F16 176 0 R /F35 178 0 R /F36 179 0 R /F8 180 0 R >> Exercise 2.12. ~WP) 6. Analytical reasoning – Video lessons. "From a Quantum Metalanguage to the Logic of Qubits" by Paola Zizzi on Arxiv.org has a couple of chapters on natural deduction systems and several references. Consider the natural deduction proof given below. 2 What it is not for; 3. /Type /Annot (Using this pack) I do not understand the step in line 10. /Subtype /Link Testing whether a proposition is a tautology by testing every possible truth assignment is expensive—there are exponentially many. 146 0 obj << Exercise 2 (Propositional logic: natural deduction (max 2 marks). 1 1. 173 0 obj << (CP) ( PW (-CO-P). /Rect [147.716 369.766 226.034 380.504] >> endobj 3. 3 Natural deduction. >> endobj Ng�;�v䒁1����e-0�kL�z(B ����dh�AgWyiϐޘ����Zr*D stream Natural Deduction. /Border[0 0 0]/H/I/C[1 0 0] /Rect [147.716 168.521 258.246 179.369] 52 0 obj /Type /Annot A → C ` (A ∧ B) → C 1 A→C 2 A∧B Ass. /Subtype /Link The pack hopefully o ers more questions to practice with than any student should need, but the sheer number of problems in the pack can be daunting. /Type /Annot /Type /Annot >> endobj endobj 24 0 obj /Border[0 0 0]/H/I/C[1 0 0] CVW A-Z 4. << /S /GoTo /D (subsection.5.7) >> >> endobj 104 0 obj 3 Natural deduction. << /S /GoTo /D (section.1) >> ABOUT; FIND THE ANSWERS. 158 0 obj << /Border[0 0 0]/H/I/C[1 0 0] (CW) 2. 108 0 obj << /S /GoTo /D (subsection.5.4) >> /Rect [147.716 144.61 206.939 155.459] /Border[0 0 0]/H/I/C[1 0 0] /Rect [466.521 276.062 478.476 284.475] : ‘ ¬ (A ∧ ¬ A). /A << /S /GoTo /D (subsection.3.1) >> 7.4 Aplia Assignment X 1. /Subtype /Link << /S /GoTo /D (section.4) >> NC V NW 4. >> endobj /Rect [470.755 475.315 478.476 483.728] endobj /Annots [ 115 0 R 116 0 R 117 0 R 118 0 R 119 0 R 120 0 R 121 0 R 122 0 R 123 0 R 124 0 R 125 0 R 126 0 R 127 0 R 128 0 R 129 0 R 130 0 R 131 0 R 132 0 R 133 0 R 134 0 R 135 0 R 136 0 R 137 0 R 138 0 R 139 0 R 140 0 R 141 0 R 142 0 R 143 0 R 144 0 R 145 0 R 146 0 R 147 0 R 148 0 R 149 0 R 150 0 R 151 0 R 152 0 R 153 0 R 154 0 R 155 0 R 156 0 R 157 0 R 158 0 R 159 0 R 160 0 R 161 0 R 162 0 R 163 0 R 164 0 R 165 0 R 166 0 R 167 0 R 168 0 R 169 0 R 170 0 R ] /Rect [466.521 194.368 478.476 202.781] /Rect [466.521 182.413 478.476 190.826] /A << /S /GoTo /D (subsection.5.5) >> Answer this question. CENGAGE MINDTAP a Search this course ? A natural deduction proof starts with a set of premises and applies introduction and elimination rules to arrive at the conclusion. 3 A ∧E 2 4 C → E 1, 3 5 (A ∧ B) → C Ben Study Resources The form of the above example should look somewhat familiar. I myself needed to study it before the exam, but couldn’t find anything useful /Type /Annot �/7t��|���iq甦�N�����UD`"��JD8�o�VtZ\ۇ�N#�M�7e�J�\{��I��xC��s}-���OF%�Uج�2 �4 /A << /S /GoTo /D (subsection.5.4) >> ( PW) (COP). /Rect [147.716 204.386 222.63 215.124] 144 0 obj << 89 0 obj Evolution: Natural selection and human selection article (Opens a modal) Artificial selection and domestication (Opens a modal) Practice. /Border[0 0 0]/H/I/C[1 0 0] View natural deduction practice problem answers.pdf from PHIL 0070 at New York University. /Border[0 0 0]/H/I/C[1 0 0] 92 0 obj /A << /S /GoTo /D (subsection.5.3) >> 1 What it is for; 3. Next lesson. 171 0 obj << 4 The derivation rules. /A << /S /GoTo /D (subsection.4.4) >>
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