by direct reasoning showing that if v is any valuation map such that v*(F) = 1 for every premise F, then v*(G) = 1 where G is the conclusion.

Question 4 Show that following. Do this without writing out truth-tables, but by direct reasoning showing that if v is any valuation map such that v*(F) = 1 for every premise F, then v*(G) = 1 where G is the conclusion. 2 marks

R/^Q

:.PKR

Question 5 Prove that if F is a propositional formula and F does not contain the symbol then F contains an odd number of symbols. (Hint: use induction on the definition of proposi-

tional formulae.)

3 marks

Question 6 Let be the set of all propositional formulae and let = be the relation of truth equivalence between formulae.

(i) Prove that = is an equivalence relation on &.

2 marks

(ii) Now assume that is the set of propositional formulae that only use the propositional variables P and Q. How many equivalence classes does & have? Explain your answer.

2 marks

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