Suppose
∣A∣=m
. What is the size of this set:
∣{X∈P(A)∈:∣X∣≤1}∣
Let's start with a specific example:
∣{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}}
Notice that the number of elements of size
≤1
is the size of the original set + 1 more element: the empty set.
Thus:
∣{X∈P(A)∈:∣X∣≤1}∣=m+1
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