Samuel Kendrick Posted on Jul 16, 2020 1 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|A|=m∣A∣=m . What is the size of this set: ∣{X∈P(A)∈:∣X∣≤1}∣ | \{X \in \mathscr{P}(A) \in : |X| \leq 1 \} | ∣{X∈P(A)∈:∣X∣≤1}∣ Let's start with a specific example: ∣{X∈P({1,2,3})∈:∣X∣≤1}∣ | \{X \in \mathscr{P}(\{1,2,3\}) \in : |X| \leq 1 \} | ∣{X∈P({1,2,3})∈:∣X∣≤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\} \} P({1,2,3})={∅,{1},{2},{3}} Notice that the number of elements of size ≤1\leq 1≤1 is the size of the original set + 1 more element: the empty set. Thus: ∣{X∈P(A)∈:∣X∣≤1}∣=m+1 | \{X \in \mathscr{P}(A) \in : |X| \leq 1 \} | = m+1 ∣{X∈P(A)∈:∣X∣≤1}∣=m+1 Top comments (0) Subscribe Personal Trusted User Create template Templates let you quickly answer FAQs or store snippets for re-use. Submit Preview Dismiss Code of Conduct • Report abuse Are you sure you want to hide this comment? It will become hidden in your post, but will still be visible via the comment's permalink. Hide child comments as well Confirm For further actions, you may consider blocking this person and/or reporting abuse
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