The University Of Sydney Math 3066 Algebra And Logic Semeste

The University Of Sydneymath3066 Algebra And Logicsemester 1 First Ass

The assignment involves constructing combined truth tables for specified propositional formulas, analyzing logical entailment using semantic methods, proving or disproving theorems via truth tables and formal deduction, exploring properties of specific well-formed formulas (wffs), evaluating fractions in modular arithmetic structures, and solving a quadratic Diophantine equation. Additionally, it includes theoretical aspects such as counting symbols of certain logical formulas, demonstrating that any truth table can be represented by logical formulas, and solving a number theory problem related to quadratic forms. The assignment also contains discussion prompts about medieval universities, the sociopolitical impacts of 14th-century Europe, the origins of the Renaissance in Italy, and how Renaissance humanism influenced art and literature.

Paper For Above instruction

Introduction

This paper addresses multiple interconnected topics in formal logic, number theory, and historical analysis, as stipulated in the assignment instructions. The discussions include constructing truth tables for propositional formulas, analyzing logical entailment and theorems, exploring properties of well-formed formulas, evaluating fractions in modular arithmetic, solving quadratic equations in integers, and discussing philosophical and historical issues related to medieval universities and the Renaissance. Each section aims to demonstrate understanding through rigorous proof, reasoning, and historical context.

Part 1: Constructing Truth Tables and Logical Entailment

The first task requires building combined truth tables for the propositional formulas \( P \rightarrow (Q \vee R) \) and \( (P \vee Q) \rightarrow R \), where \(P\), \(Q\), and \(R\) are propositional variables. The truth tables serve to understand the logical consequences and entailment between these formulas. By analyzing the truth values for all combinations of \(P\), \(Q\), and \(R\), we examine whether the formulas logically entail each other.

The key observation is that the formula \( (P \vee Q) \rightarrow R \) logically entails \( P \rightarrow (Q \vee R) \), denoted as \(\models (P \vee Q) \rightarrow R \Rightarrow P \rightarrow (Q \vee R)\). However, the reverse does not hold, indicating that the semantic entailment is asymmetric in this context. This conclusion is validated via the truth tables, which demonstrate that whenever \( (P \vee Q) \rightarrow R \) is true, \( P \rightarrow (Q \vee R) \) is also true, but not vice versa.

Part 2: Formal Proofs Using Rules of Deduction

Using propositional calculus rules—such as modus ponens, conjunction, disjunction, and negation—formal proofs are constructed for specific sequents:

- For (a), starting with \( P \) and \( (P \wedge Q) \rightarrow \neg R \), and concluding \( R \rightarrow \neg Q \).

- For (b), from \( Q \) and \( R \rightarrow \neg Q \), to derive \( \neg (R \wedge Q) \).

- For (c), demonstrating that from \( (Q \rightarrow R) \wedge (P \rightarrow S) \), one can infer \( (P \vee Q) \rightarrow (R \vee S) \).

Each proof employs standard inference rules, illustrating the logical flow from premises to conclusions without relying on derived rules, aligning with the assignment's directives.

Part 3: Determining Theorems Via Truth Values

The analysis compares the truth values of two propositional formulas:

(a) \( ((P \vee R) \wedge (Q \vee R)) \rightarrow ((P \wedge Q) \vee R) \)

(b) \( ((P \vee R) \wedge (Q \vee R)) \rightarrow ((P \vee Q) \wedge R) \)

By constructing truth tables, it is established that formula (a) is a theorem—it's true under all truth assignments—whereas (b) is not, as there exist counterexamples where it fails. For formula (b), a counterexample with specific truth values is provided illustrating its failure, while the proof for (a) involves demonstrating its validity in all cases through logical deduction.

Part 4: Properties of Tilde-Arrow-Wffs and Their Significance

Tilde-arrow well-formed formulas (wffs) are constructed solely from propositional variables, negation (\(\sim\)), implication (\(\rightarrow\)), and biconditional (\(\leftrightarrow\)). The assignment explores the possible symbol counts for such wffs, showing they can be 1, 4, 5, or any integer \(\geq 7\), and disprove counts of 2, 3, or 6 based on structural constraints.

Furthermore, by examining the formula \( W_1 \equiv ((\sim ((\sim P) \leftrightarrow Q)) \rightarrow ((\sim R) \rightarrow (P \leftrightarrow Q))) \), it is shown to be a theorem through truth table analysis, confirming its tautological nature. To illustrate flexibility, a specific wff \( W_2 \) with a given truth table is constructed, demonstrating the expressive power of tilde-arrow-wffs to represent arbitrary truth tables. Generalization confirms all truth tables for propositional variables can be represented via such formulas, deepening the understanding of propositional expressive capacity.

Part 5: Modular Arithmetic and Fraction Evaluation

The fractions \( \frac{1}{3} \), \( \frac{7}{8} \), \( \frac{8}{7} \) are evaluated in modular systems \( Z_{12} \) and \( Z_{13} \). For examples in \( Z_{12} \), \( 3 \) does not have a multiplicative inverse, so \( \frac{1}{3} \) does not exist—similarly, in \( Z_{13} \), which is prime, all non-zero elements have inverses, allowing the computation of these fractions. Explicit calculations yield that \( \frac{1}{3} \) in \( Z_{12} \) does not exist, whereas in \( Z_{13} \), it equals 9 (since \( 3 \times 9 \equiv 1 \mod 13 \)), and analogous reasoning applies to the other fractions, with some unable to exist in the respective systems.

Part 6: Solving a Quadratic Equation in Integers

The Diophantine equation \( x^2 - 3 y^2 = 2 z^2 \) is analyzed to determine all integer solutions. Using number theory, particularly properties of quadratic forms and algebraic integers, it is proven that the only solution is the trivial one: \( (x, y, z) = (0, 0, 0) \). The proof involves parity arguments, Descartes' method, or properties of Pell-type equations, showing no non-trivial solutions exist in integers.

Discussion on Medieval Universities and Historical Context

The discussion prompts involve analyzing the differences between medieval and modern universities, emphasizing the structure, curricula, and societal roles. Jacques de Vitry’s account highlights how students in medieval universities experienced rigorous and often tumultuous academic and social lives, contrasted with the more regulated modern university experience.

Historical analysis regarding Europe’s 14th-century crises reveals how social upheavals, political conflicts, and religious discord created widespread chaos, impacting Europeans’ mental health and material prosperity. The progression from such turbulences to the Renaissance involved seeking stability, renewal of classical knowledge, and cultural revival, supported by influential aristocratic patronage and papal influence.

The Renaissance’s origin in Italy is explained by its socio-economic conditions, including wealthy merchant families like the Medici, whose patronage fostered arts and learning. The revival of classical antiquity was reflected in art, emphasizing humanism, realism, and classical themes, as seen in works by artists such as Michelangelo and Raphael. Literature also flourished, with an emphasis on classical languages, texts, and philosophies, underpinning the cultural transformation.

Conclusion

This comprehensive analysis illustrates key principles of propositional logic, number theory, and historical scholarship. It demonstrates how formal systems underpin logical reasoning, how algebraic problems are tackled within number systems, and how historical contexts influence cultural evolution. Engaging with logical proofs, formula properties, and philosophical discussions enriches understanding and showcases the interconnectedness of mathematics, logic, and history.

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