Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof/7/b/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 7 / b / Solution by Codex 0 2026-10-25
Apply the complete logical derivation of the law of excluded middle in (a) to , obtaining . Then use the disjunction elimination rule, constructing each logical implication by an explicit hypothetical subproof:
1  Q ∨ ¬Q                              result of (a), φ := Q
2  | Q                                 first case
3  | | P                               assume
4  | | Q                               available from 2
5  | P → Q                             →-intro, discharge 3–4
6  | (P → Q) ∨ (P → ¬Q)                ∨-intro-L, 5
7  | ¬Q                                second case
8  | | P                               assume
9  | | ¬Q                              available from 7
10 | P → ¬Q                            →-intro, discharge 8–9
11 | (P → Q) ∨ (P → ¬Q)                ∨-intro-R, 10
12 (P → Q) ∨ (P → ¬Q)                  ∨-elim, 1, cases 2–6 and 7–11
Both case assumptions and both hypothetical assumptions have been discharged. Thus the conclusion is a theorem, not a logical implication that depends on assuming .
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