Question

Part 3 (one point total). For each of the following sequents, provide a proof that demonstrates their validity. Make sure to use SL and the primitive rules (no derived rules). 1. AB,BC FAC 2. A->B, BC, DEA&D-E&C 3. -AVB, -BVC, -DVEA&DE&C 4. -AVB, -DVEF (A-B)&(D–>E) 5. ( AB)+((B+C)&( DE), A +-AVBA&D>E&B 6. P→Q-QT-P 7. P Q –QF -P 8. P Q -PF-Q 9. P&PTD 10. P-Q, Q--PF P™D

Answer 1

In logic, we have to be clear about the meaning of each symbol. We should know how to read the given sequent in order to prove it.

'→' read as 'implies'. Meaning of A→B is if A is true then B is also true.

'⊢' read as 'then'. Meaning is if all of left happens then it will also happen all of right.

Please find the attached images for the proof of all given sequents.
(10) P7Q , Q NP FP D. roved 1 PAQ (2) o ~P (3) P 2,4 (5) ~P this leads to P ~P. (6) P (7) ONP 6,7 (8) U (universal set) tuis leads to Ро & PD >P>Q, & aupf PD (9) (1P (2) ~P 4,2 (3) U (universal set) this leads to pyu P>D P&~PEDI

0 AA, BLANC 1 AB (2) BC (3) A (4) 3 (5) 1,9 2,4 * AB, BAC, AAC this leads to Арес ЕАЭС Proved E&C A B B ₃C, D EFA&D (1) AB (2) BC (3) A 1, 3 (4) B 2, 4 (5) this leads to AB, B > CAC 1, 2 6,7 (6) DE (7) A C (8) A & D E & C this leads to AB, BC, DEFA& DE&C proved

(5) (AB) (BC) & ( DE)), A NAVBt A & D E&B (1) ( A B) (B+C) & (DE) Assume Assume 9,215) B C 12 (6) (2) D 6, 7 (8) E (9) A 210) B (12) A&D 11 (12) B &E Assune E-617 Assume - E2 Assume (13) A NA VB (14) A Assume lis) NA VB holds true 13, 14, 15/16) B 14, 16 (17) A B tuis leads to ( AB) ((B3) 4D 4EJ), A MNAVO AfD-MERA

(4) NAVB, ND NE I ( AB) & (DE) (1) ~AVB I NAVO holds true - A = true wp= false = (2) A =true 123) (4) ~DIE (5) D I same logic 4,5(6) E 2, 3 (7) A 5,6 (8) D B E this leads to NAVB ,~D VE (AB) & (DE) v proved

(3) NAVB, Novc, NDVEH A&D EXC Loved I WA VB holds & A true => ~ A Gfalse y Big true B is true & ~B = false c es true W~AVB (2) A ,2 (3) B I (4) ~BVC 3,4 (5) c 1,2,4 (6) A C NOVE (8) D 7,8 (9) E 7,8,9 (10) D E INDVE holds & D-true »ND = false & E=true this leads to INA VB, ~B VC , ND VE FA&DE &C. proved

(6) Po.-at wp (2) (3) P (4) Q E12 3,4,5 (6)~P D . this leads to P>0,- Proved (7) PEQ, ~Q t ap (3) No (4) P 4,4 (5) 3,4,516) ~P this leads to PAO, NI (8) Peoinpt ~a (1) PD (2) OP (3) ~p (4) IP 1,4 (5)10 this leads to Pesa, apt wat 3,4,516) no