Let P be some predicate. Check the box next to each scenario in which n N, P(n) must be true.

a) For every natural number k > 0 , if P(i) holds for every natural number i < k, then P(k) holds.

b) P(0) holds and for every natural number k > 0, if P(i) does not hold, then there is some natural number i < k such that P(i) does not hold.

c) For every natural number k, if P(i) holds for every natural number i < k, then P(k) holds.

d) For every natural number k, if P(k) does not hold, then there is a smaller natural number i < k such that P(i) does not hold.

SELECT ALL THAT APPLY.