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2. For the sequents below, show which ones are valid and which ones aren't: * (a) -p -"qq→P (b) -p v-q -(pnq) * (c) "p, P vq9

2. For the sequents below, show which ones are valid and which ones aren't: * (a) -p -"qq→P (b) -p v-q -(pnq) * (c) "p, P vq9

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. For the sequents below, show which ones are valid and which ones aren't: * (a) -p - -99→P (bv d)- b A d (q) * * (c) p, p vqq (d) p v q,-q vrpvr (e) p (q vr),-q,-r-p without using the MT rule * (f) -p A -g -(p v q) * (b+4) v (b+.1)_ d v d (8), (h) pq, s→tp vs qAt *(i) -(-p v q) p. 3. Prove the validity of the sequents below: (a) p pp (b) pp 4 (c)p v q,"qp (d)-pp(p q)) (e) (p- (f) p9 P vq (g) p vq(p→4) 80 1 Propositional logic (h) p (q vr), -q, -r |- -p (i) (c An) t, h A s, hA-(s vc) →p|- (nA-t)→p (G) the two sequents implict in (1.2) on page 20 (k) q |- (p a q) v(-p^q) using LEM (1) -(p ag) |- -pv-g (m) p Aqr- (pr) v (q→r) (n) p aq -(-p v -q) (0) (-p v -q) pnq (p) p q p v q possibly without using LEM? () (p q) v (q→r) using LEM (r) p - q, -p -r, "q→ rq (s) p- q, r-t, q →rp- -t (t) (p → q) → r, s p, t, s A t→qr (u) (s → p) v (1 → q) (s → q) v (t→ p) q) qp *
Transcribed Image Text:2. For the sequents below, show which ones are valid and which ones aren't: * (a) -p - -99→P (bv d)- b A d (q) * * (c) p, p vqq (d) p v q,-q vrpvr (e) p (q vr),-q,-r-p without using the MT rule * (f) -p A -g -(p v q) * (b+4) v (b+.1)_ d v d (8), (h) pq, s→tp vs qAt *(i) -(-p v q) p. 3. Prove the validity of the sequents below: (a) p pp (b) pp 4 (c)p v q,"qp (d)-pp(p q)) (e) (p- (f) p9 P vq (g) p vq(p→4) 80 1 Propositional logic (h) p (q vr), -q, -r |- -p (i) (c An) t, h A s, hA-(s vc) →p|- (nA-t)→p (G) the two sequents implict in (1.2) on page 20 (k) q |- (p a q) v(-p^q) using LEM (1) -(p ag) |- -pv-g (m) p Aqr- (pr) v (q→r) (n) p aq -(-p v -q) (0) (-p v -q) pnq (p) p q p v q possibly without using LEM? () (p q) v (q→r) using LEM (r) p - q, -p -r, "q→ rq (s) p- q, r-t, q →rp- -t (t) (p → q) → r, s p, t, s A t→qr (u) (s → p) v (1 → q) (s → q) v (t→ p) q) qp *
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