The number of elements in a power set of size <= 1 is the size of the original set + 1 more element: the empty set .

#math #sets

Suppose $|A|=m$ . What is the size of this set:

| \{X \in \mathscr{P}(A) \in : |X| \leq 1 \} |

Let's start with a specific example:

| \{X \in \mathscr{P}(\{1,2,3\}) \in : |X| \leq 1 \} |

First, let's create the power set, stopping once we hit an element whose size is greater than 1:

{\mathscr{P}(\{1,2,3\})} = \{\emptyset, \{1\}, \{2\}, \{3\} \}

Notice that the number of elements of size $\leq 1$ is the size of the original set + 1 more element: the empty set.

Thus:

| \{X \in \mathscr{P}(A) \in : |X| \leq 1 \} | = m+1

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