Question

Please give proof direct or indirect with numbered justification/law.

(a) t→r, ¬(r∨¬q), ¬t→p, p→(s∨¬q) ⇒ s

(b) (s→q)∧(p→t) ⇒ (s∨p)→(q∨t)

Answer 1

>S: >P, P-> (SV29) t7 (129) 2 a) t+ 7(8124 Direct Method Step Reason Premise 1. ZtP contrapositive of 2 3. p-st t* 4. zp yr 5. lov zq) 6. zrna 7 ?8. Premise chain rule on @ @ Premise De-Morgan's law on ☺ conjunctive simplification of Models tollens on © ¢ ® o it so BT o SV 29 p>(SV 29) premise Modus ponens on ® 4 9->S conditional equivalence 12 9. conjunctive simplification on Modus poners on ④ +12 Hence given argument is valid. (b)(s~9)^(p->t) → csup) → cqvt) Indirect Method, (5-79) 1.0p->t), z (csup)-> (9v+)) + F Steps Reason z(esvp) →Cqv+)) Premise coisvp vieAvt)) conditional equivalence of O CamScanner

CSVP) A?(9vt) De Morgans law on @ Conjunch've simplification SVP 6 ($*&A (prt) Premise @ ⓇO 0 00 00 00 00 00 Pt conjuctive simplification to Disjucive simplification 4 Models ponens ont rqv) conjunctive simplification et zqnzt De Morgan's law on © et conjective simplificn cm s>q conjuctive simplification ons Disjuctive simplifica 7 © modes ponens on B 3 conjunctive simplifich on qazq conjunction of D9 - pazpf conjuction 076 +0 • PAPER FAR conjunction of © ¢ ® PAP =P Hence the given argument is valid that Scanned with CamScanner