Final answer:
By using strong induction, we prove that any natural number ≥ 12 can be represented using sets of 4 or 5 by verifying base cases and assuming the proposition for all natural numbers up to k to show it holds for k+1.
Step-by-step explanation:
To prove the statement that any natural number greater than or equal to 12 can be made using sets of 4 or 5, we use strong induction as our method. Strong induction requires that we assume our proposition is true for all natural numbers less than or equal to a certain value and then use this to show that it must also be true for the next natural number.Firstly, we must verify the base cases, which in this scenario are 12, 13, 14, and 15. These can be verified easily:
- 12 can be made by using three sets of 4 (4+4+4)
- 13 can be made by using two sets of 4 and one set of 5 (4+4+5)
- 14 can be made by using one set of 4 and two sets of 5 (4+5+5)
- 15 can be made by using three sets of 5 (5+5+5)
Now, assume that for some k ≥ 12, every number greater than or equal to 12 and less than or equal to k can be formed using sets of 4 or 5. We need to prove that k+1 can also be formed using these sets. Since k ≥ 12, k – 3 ≥ 9, and by our assumption, k – 3 can be formed using sets of 4 or 5. Therefore, we can simply add one more set of 4 to k – 3 to get k+1 (since (k – 3) + 4 = k + 1).This shows that any number greater than or equal to 12 can indeed be constructed using combinations of the set of 4 or the set of 5, thus completing the proof via induction.