Question

Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible. \(\mathbb{N} \notin \mathscr{P}(\mathbb{N})\)

Short answer

\(\mathbb{N} \notin \mathscr{P}(\mathbb{N})\) is a statement, and it is false because the set of natural numbers is a subset of its own power set.

Step-by-step solution

Understanding the concepts of set and power set

Before we can evaluate the statement, we need to understand what a set and a power set are. A set is a collection of distinct objects, in this case, the set of natural numbers \(\mathbb{N}\). The power set of a set is the set of all possible subsets of the set. In this case, the power set of \(\mathbb{N}\) is denoted by \(\mathscr{P}(\mathbb{N})\).

Identifying if the set is a subset of its power set

By definition, the power set of any set includes the set itself as one of its elements. So, \(\mathbb{N}\) is an element of \(\mathscr{P}(\mathbb{N})\), that is \(\mathbb{N} \in \mathscr{P}(\mathbb{N})\). Therefore, the statement \(\mathbb{N} \notin \mathscr{P}(\mathbb{N})\) is false.

Final conclusion

Since the statement \(\mathbb{N} \notin \mathscr{P}(\mathbb{N})\) can be assigned a truth value (that is, it is false), we can conclude that it is indeed a statement, but a false one.