The set of strings of finite length formed from the symbols and is denoted or . A system of deduction for strings is given with deduction rules
We write to mean may be deduced from by finitely many applications of these rules.
(a) Find the set , carefully justifying your answer.
(b) Let be the number of s in plus twice the numbers of s in . Show that implies . Deduce that .
(c) Prove that is finite for all .
(d) Explain why (b) above may be regarded as
soundness theorem. Briefly suggest a corresponding
completeness theorem and state one or more additional
deduction rules that may be required for your completeness
theorem to be true.
(a) Let be a proposition with truth value true or false. Show that one of
is a true statement irrespective of whether or not is true or false and the other is not.
(b) A proof system is presented for the alphabet with symbols , , , , but no brackets. A well-formed-formula (wff) is a string of the form , where are arbitrary letters (not necessarily distinct) and there are one or more such letters. The rules for this proof-system are:
for any wffs . Show that if can be derived in then contains an odd number of occurences of the symbol .
(c) Give an example of a string of symbols from which has an odd number of occurrences of but is not derivable in . Hint: consider the number of times each letter can occur in the string .