Consider the following propositional symbols and their intended meanings p The shop is open today 4 Sarah is going to the shop today r John is going to the shop today (1) Express the following declarative sentence in propositional logic using the propositional symbols as given above The shop is open today, and either John or Sarah or both are going (2) (11) Express the following propositional logic formula in English where the propositional symbols have the meanings given above (2) (p) AT Question 1.2 Use the basic natural deduction rules for propositional logic to prove the validity of the following sequents 1 (5) F (pvq) + (pq) 11 pF(qa-p) v (p^-4) Question 1.3 Show that the following sequent is not valid by giving an appropriate valuation (4) (p^ q) r7(p->r) ^ (9-) Explain why your valuation proves that the sequent is not valid [TURN OVER] 3 COS3761 May/June 2019 Question 1.4 Use the HORN algorithm to prove that the following Horn formula is satisfiable" or "unsatisfiable" Show each step (p^q^w+ 1)^(t+1)^ (r + p) ^ (T+r)^(T q) ^ (r^uw) ^ ( us)