解决这道题---Problem 2. (12 points) We used case analysis in some of our proofs in class, and Shaddin mentioned that this is justified by the rules of logic. In this problem, we will make this justification more precise. (a) [2 points]. Recall problem 6c from Hw1. Explain how this justifies a case analysis with two cases. (b) [5 points]. Using induction, show that the following holds in propositional logic for any constant k. p₁ ⇒ r p₂ ⇒ r ... p_k ⇒ r _________ ∴ (p₁ ∨ p₂ ∨ ... ∨ p_k) ⇒ r Explain how this justifies case analyses with any finite number of cases. (c) [5 points]. Let P(x) and R be formulas in first order logic, and suppose that the variable x does not appear in R. Show, using the rules of inference of first order logic presented in class, that ∀x(P(x) ⇒ R) entails (∃xP(x)) ⇒ R. Your (short) proof should use no more than four applications of the inference rules from class. Explain how this justifies a case analysis with any number of cases (possibly infinite).

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Part (a) asks us to recall Problem 6c from Homework 1, which likely involved proving that the formula ((P implies Q) and (not P implies Q)) implies Q is a tautology. This formula justifies case analysis with two cases. The Law of Excluded Middle tells us that either P or not P must be true. If we can prove that Q follows from P, and also prove that Q follows from not P, then Q must be true regardless of which case holds. This is the logical foundation for two-case proof by cases. Part (b) asks us to prove by induction that if we have k implications from p₁ to r, p₂ to r, up to pₖ to r, then we can conclude that the disjunction of all p's implies r. The base case for k equals 1 is trivial. For the inductive step, we assume the statement holds for k and prove it for k plus 1. We use disjunction elimination: if we assume the disjunction of all cases holds, then either the first k cases hold or the k plus 1 case holds. In either situation, we can derive r. This justifies case analysis with any finite number of cases, because if the cases are exhaustive and each case implies our conclusion, then the conclusion must be true. Part (c) asks us to show that for all x, P of x implies R, entails the existence of x such that P of x, implies R, where x is not free in R. We prove this using exactly four inference rules. First, we assume the existential statement. Then we use existential elimination to assume P of c for some arbitrary constant c. We apply universal elimination to get P of c implies R. By modus ponens, we derive R. We discharge the assumptions using existential elimination and conditional proof. This justifies case analysis with infinitely many cases, because if we know that R follows from P of x for any x, and we know there exists at least one x satisfying P, then R must be true regardless of how many such x exist. This visual summary shows how case analysis is justified across different logical systems. For two cases, we rely on the Law of Excluded Middle that either P or not P must be true. For finite cases, our induction proof demonstrates that if we can prove our conclusion in each exhaustive case, then the conclusion holds overall. For infinite cases, first-order logic allows us to reason about arbitrary elements in potentially infinite domains. All three forms of case analysis are rigorously justified by formal inference rules, making case analysis a fundamental and reliable proof technique in mathematics and logic. In conclusion, we have completely solved Problem 2 by demonstrating how case analysis is justified in logic. Part (a) showed that two-case analysis relies on the tautology that if both P implies Q and not P implies Q, then Q must be true. Part (b) used mathematical induction to prove that finite case analysis works for any number of cases. Part (c) demonstrated that infinite case analysis is justified through first-order logic inference rules. The key insight is that case analysis, whether involving two cases, finitely many cases, or infinitely many cases, is fundamentally grounded in rigorous logical principles. This makes case analysis a reliable and powerful proof technique across all areas of mathematics and logic.