Question
Problem Set 3 (PHI 251) Due Fri. February 9th by 5 pm Eastern Please scan and upload to Blackboard as a pdf; no pictures allowed;
Problem Set 3 (PHI 251)\ Due Fri. February 9th by 5 pm Eastern\ Please scan and upload to Blackboard as a pdf; no pictures allowed; feel free\ to also turn in a paper copy to Philosophy Dept, Hall of Languages Room 541\ For EACH problem, label the base case(s) AND label the induction step/hypothesis.\ Call a string over
{a,b}
an "a-palindrome" if it is a palindrome that has "a" as a\ middle letter. (An a-palindrome therefore must have an odd number of letters.)\ (i) Give a recursive definition of the set of "a-palindromes", and\ (ii) prove by induction that every a-palindrome has an even number of "
b
"'s.\ Prove that no well-formed formula of sentential logic ever contains consecutive\ atomic formulas [e.g. nothing like '
(PP&Q)
'.]\ Complete two of the remaining three problems: you get to pick your favorite two!\ Here is the recursive definition of
n
! (read "
n
factorial"):\
1!=1\ (n+1)!=(n+1)\\\\times n!
\ That is,
n!=ubrace((n\\\\times (n-1)\\\\times dots3\\\\times 2\\\\times 1)ubrace)_(n times )
\ Prove by induction: For every
n
greater than or equal to
a,b,c,dn2
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