Question

On a far away galaxy, Aliens developed a (weird) limited language using only the words "good", "mood" and "food". They make new words by concatenating (juxtaposing) these words in any order. For example "foodgoodfood" is a valid word in their language. Any word produced like this is a correct word in their language. Prove using strong induction that any word in this language has an even number of o's. (Hint: Use strong induction on the number of sub-words in the word).

Answer 1

"They are making words from these only words ""good, food, mood: concatenating in Any order. So words have only these Sub - words ""good, food, mood"" proof p (n): Suppose in Any word of aliens language produced by concatenating N Sub - words then in that word the number of 0's is even forall n x 1 p (n) = 2n (Number of 0's) proof of strong Interaction Case 1: P (1): ""mood"" or ""food"" OR ""good"" it has 20's which is an (Tre) P (2): ""moodfood"" it hp 40's which i] even (true) P (3): ""food mood good"". it hp 60's which i] even (True) Case 2: Assume that P (n) is True forall alpha lessthanorequalto n lessthanorequalto k to prove p (k + 1) = 2 (k + 1) p (k + 1) = p (k) + 2 = 2k + 2 which pi even we have p (k) is True p (k + 1) pi 2 (k + 1) is "