Beginning Activity Beginning Activity 2: A Property of the Natural Numbers
Intuitively, the natural numbers begin with the number 1, and then there is 2, then 3, then 4, and so on. Does this process of “starting with 1” and “adding 1 repeatedly” result in all the natural numbers? We will use the concept of an inductive set to explore this idea in this activity.
Definition.
A set \(T\) that is a subset of \(\mathbb{Z}\) is an inductive set provided that for each integer \(k\text{,}\) if \(k \in T\text{,}\) then \(k + 1 \in T\text{.}\)
1.
Carefully explain what it means to say that a subset \(T\) of the integers \(\Z\) is not an inductive set. This description should use an existential quantifier.
2.
Use the definition of an inductive set to determine which of the following sets are inductive sets and which are not. Do not worry about formal proofs, but if a set is not inductive, be sure to provide a specific counterexample that proves it is not inductive.
(a)
\(A = \left\{ {1,2,3, \ldots ,20} \right\}\)
(b)
The set of natural numbers, \(\mathbb{N}\)
(c)
\(B = \left\{ { {n \in \mathbb{N}} \mid n \geq 5} \right\}\)
(d)
\(S = \left\{ { {n \in \mathbb{Z}} \mid n \geq - 3} \right\}\)
(e)
\(R = \left\{ { {n \in \mathbb{Z}} \mid n \leq 100} \right\}\)
(f)
The set of integers, \(\mathbb{Z}\)
(g)
The set of odd natural numbers.
3.
This part will explore one of the underlying mathematical ideas for a proof by induction. Assume that \(T \subseteq \mathbb{N}\) and assume that \(1 \in T\) and that \(T\) is an inductive set. Use the definition of an inductive set to answer each of the following:
(a)
Is \(2 \in T\text{?}\) Explain.
(b)
Is \(3 \in T\text{?}\) Explain.
(c)
Is \(4 \in T\text{?}\) Explain.
(d)
Is \(100 \in T\text{?}\) Explain.
(e)
Do you think that \(T = \mathbb{N}\text{?}\) Explain.