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

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