$65.$ An obscure tribe has only three words in its language, moon, noon, and soon. New words

are composed by juxtaposing these words in any order, as in soonnoonmoonnoon. Any

such juxtaposition is a legal word.

a$.$ Use the first principle of induction $($on the number of subwords in the word$)$ to prove

that any word in this language has an even number of o$\text{'}$s$.$

b$.$ Use the second principle of induction $($on the number of subwords in the word$)$ to

prove that any word in this language has an even number of o$\text{'}$s$.$