i used simple induction for all the three questions. Since i guess it would be easier to prove P(n)=>P(n=1) for the predicate in the three questions, except the last question. it seems it would be proper if i used complete induction to prove it. Similar problem i had seen before. since i was writing in a hurry, i just picked the most familiar technique for me to solve, which is simple induction. I was stuck when i was dividing the set into a part that can be applied with IH. The n+1 element took me a lot of time to derive the subsets containing it. I might be wrong on the answer since the induction step wasn't that successful. if there were 5 more minutes maybe i could do the induction step one more time and find out which number of subsets is going wrong.
Luckily the first two questions required less cases to analyze. i think i did well in the recursive one.
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