6. [-/4 Points] DETAILS HUNTERDM3 3.4.022. MY NOTES ASK YOUR TEACHER Define a set X recursively as follows. B. 3 and 7 are in X. R. If x and y are in X, so is x + y. (Here it is possible that x = y.) Prove that, for every natural number n 2 12, n EX. (Hint: For the base case, show that 12, 13, and 14 are in X.) Base Case: By the base case of the definition of X, 3 and 7 are in X. Since 12 = (3 + 3) + (3 + ). 12 E X by the recursive case of the definition. Since 13 = (3 + 3) + 13 EX. And since 14 = 7 + , 14 EX. Inductive Hypothesis: Suppose as inductive hypothesis that i G X for all i such that 12 s i 14. Inductive Step: k - 3 CX by inductive hypothesis. By the recursive part of the definition, + 3 = kEX, as required. eBook Submit