1. Let ø be an SL-formula. Prove that o is logically equivalent to -nø. 2. Prove that {¢,v} E ¢ ^ ½. 3. Prove both (a) {¢ ^ b} E ¢ and (b) {ø ^ v} E b. 4. Prove that Ø E ((4 Ɔ v) Ɔ ø)ɔ. 5. Prove that the following sentential axioms are tautologies: (1) ø Ɔ (4 Ɔ $), (2) (ø Ɔ (ý Ɔ x)) Ɔ (($ɔ ) > ($ Ɔ x), (3) (¬ø ) ¬ab) Ɔ (½ ɔ ¢). 6. (Bonus question) Recall definitions 6 and 7 in the Proof Theory section of the course notes (currently p. 27). Try to prove {ø Ɔ v, ¬x } FØƆ x. Hint: Use the proof of {$ Ɔ ½, V Ɔ x} E¢Ɔ x from p. 29.
1. Let ø be an SL-formula. Prove that o is logically equivalent to -nø. 2. Prove that {¢,v} E ¢ ^ ½. 3. Prove both (a) {¢ ^ b} E ¢ and (b) {ø ^ v} E b. 4. Prove that Ø E ((4 Ɔ v) Ɔ ø)ɔ. 5. Prove that the following sentential axioms are tautologies: (1) ø Ɔ (4 Ɔ $), (2) (ø Ɔ (ý Ɔ x)) Ɔ (($ɔ ) > ($ Ɔ x), (3) (¬ø ) ¬ab) Ɔ (½ ɔ ¢). 6. (Bonus question) Recall definitions 6 and 7 in the Proof Theory section of the course notes (currently p. 27). Try to prove {ø Ɔ v, ¬x } FØƆ x. Hint: Use the proof of {$ Ɔ ½, V Ɔ x} E¢Ɔ x from p. 29.
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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Transcribed Image Text:1. Let ø be an SL-formula. Prove that o is logically equivalent to -nø.
2. Prove that {¢,v} E ¢ ^ ½.
3. Prove both (a) {¢ ^ b} E ¢ and (b) {ø ^ v} E b.
4. Prove that Ø E ((4 Ɔ v) Ɔ ø)ɔ.
5. Prove that the following sentential axioms are tautologies:
(1) ø Ɔ (4 Ɔ $),
(2) (ø Ɔ (ý Ɔ x)) Ɔ (($ɔ ) > ($ Ɔ x),
(3) (¬ø ) ¬ab) Ɔ (½ ɔ ¢).
6. (Bonus question) Recall definitions 6 and 7 in the Proof Theory section of the course
notes (currently p. 27). Try to prove {ø Ɔ v, ¬x } FØƆ x. Hint: Use the proof of
{$ Ɔ ½, V Ɔ x} E¢Ɔ x from p. 29.
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