Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2018/digital-systems-linear-algebra-introduction-to-formal-proof/7/d/ii/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Digital systems, linear algebra, introduction to formal proof / 7 / d / ii / Solution by Codex 0 2026-10-31
Use the exclusive-or natural deduction rules defined in part (i). The only extra lemma needed is double-negation introduction, derived from the given rules by assuming under , obtaining contradiction, and discharging that assumption. The full proof incorporates that derivation:
1. ¬P ⊕ Q               premise
2.   ¬P                 first ⊕E case assumption [i]
3.   ¬Q                 first ⊕E case assumption [j]
4.   P ⊕ ¬Q             ⊕I₂, 2, 3
5.   ¬¬P                second ⊕E case assumption [k]
6.   Q                  second ⊕E case assumption [l]
7.   P                  ¬¬E, 5
8.     ¬Q               assumption [m]
9.     ⊥                ¬E, 6, 8
10.  ¬¬Q                ¬I, 8–9, discharging [m]
11.  P ⊕ ¬Q             ⊕I₁, 7, 10
12. P ⊕ ¬Q              ⊕E, 1, 2–4, 5–11,
                        discharging [i], [j], [k], [l]
In the first case the second introduction rule is instantiated with right operand . In the second case the first introduction rule requires , which is exactly the derived line 10.
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