Inductive Sets Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

The notions of “inductive set,” and of how to prove that a set is inductive, underlie proof by induction. This discussion helps ensure that these ideas make sense to you.

Discuss which of the following statements are true and which are false, and why:

1. The set of integers greater than or equal to 0 is inductive.
2. The set of integers less than or equal to 0 is inductive.
3. If A is an inductive set, then 1 ∈ A.
4. If A is an inductive set, and 1 ∈ A, then 100 ∈ A.
5. The set of positive integer multiples of 1/2 (i.e., {1/2, 1, 3/2, 2, 5/2, ...}) is inductive.
6. The set of numbers computed by multiplying 1/2 by an odd positive integer, i.e., {1/2, 3/2, 5/2, ...}, is inductive.
7. If A is an inductive set, and A ⊆ B, then B is inductive.
8. No inductive set has a largest member.

What examples of inductive sets, other than the ones given in the textbook, can you think of?

Finally, as a glimpse of how to prove sets inductive, can you prove that the set of natural numbers n for which 4n + 3 is odd is inductive? You might want to start just by listing some members of this set, to be sure you all agree on what it is.