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1. φε Γ ND-assumption TE TO 02 ND 1 - To → ΓΕ TOND-E ΓΕφι – θα ΓΕ ό: TO TE ΓΕ οι ΛΟΣ ND-AL ΓΕ οι Λό2 ND-ΛΕΙ a) Given any propositional formulae , and dz, we already saw the introduction and elimination rules for conjunction (01 102) and disjunction (01 v 02) operations using natural deduction in the lectures, also provided in the appendix. Your task is to define the introduction and elimination rules for the xor (0.02) operation in a similar way, also using the natural deduction system of logic. [5 marks] b) Prove the following constructive logic formula using the rules of natural deduction provided in the appendix + (0 - 0) – (°' → 0) – (4 →*') → o' → *') [5 marks] c) Assume that we have a proof of P = (01 - 02) 01) ► 01 for any 01 and 02, and - ΓΕ οι TOLA 02 ND-AE2 - TE02 ΓΕ οι ND-VII ΓΕόι νο2 TE02 ND-VIZ ΓΕθινός To, Toto TH ΓΕφινόη ND-VE ND-truth TET ΓΕΙ ΓΕφ ND-explosion Q = (((øv (-º)) –1) – (øv (-º))) for any º. V Note: no is a derived construct, it is defined as on 1 Now provide a proof of the excluded middle, y v (4), in natural deduction using the assumed proof of P and Q. Activate Windows Go to Settings to activate[5 marks
1. a) Given any propositional formulae 0, and z, we already saw the introduction and elimination rules for conjunction (01 102) and disjunction (01 v 02) operations using natural deduction in the lectures, also provided in the appendix. Your task is to define the introduction and elimination rules for the xor (01 02) operation in a similar way, also using the natural deduction system of logic. [5 marks) b) Prove the following constructive logic formula using the rules of natural deduction provided in the appendix + (0-4) -(0' ) – (4 -4') → (°' ') [5 marks) c) Assume that we have a proof of P = ((01 - 02) - 01) - 0, for any 0, and oz, and Q = ((($V (-º)) –+) – (V (-º))) for any $. Now provide a proof of the excluded middle, y v (4), in natural deduction using the assumed proof of P and Q. [5 marks]
σε Γ ND-assumption ΓEό Γ,όι Fo2 ND-I ΓΕόι - ό: ΓΕφι – φ2 ΓΕόι ND-E ΓEό ΓΕόι ΓΕΣ ND-AI ΓΕφι Λό2 ΓΕφη ΑΦ2 ΝΙ-ΛΕΙ ΓΕ οι ΓΕφη Αφα ND-AE2 ΓΕ02 ΓΕ Οι ND-VI1 ΓΕόι νο2 ΓEό2 ND-V12 ΓΕφι νό2 Γ, Φι Εφ' Γ, Φ2 - φ' ND-VE ΓΕσ' ΓΕφη νό2 ND-truth ΓΕΤ ΓΕΙ ND-explosion ΓEό Note: -o is a derived construct, it is defined as 0+1