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

adnauseum profile image Samuel Kendrick ・1 min read

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

{XP(A):X1} | \{X \in \mathscr{P}(A) \in : |X| \leq 1 \} |

Let's start with a specific example:

{XP({1,2,3}):X1} | \{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:

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

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

Thus:

{XP(A):X1}=m+1 | \{X \in \mathscr{P}(A) \in : |X| \leq 1 \} | = m+1

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