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→When A is countably infinite: Added "Indeed," to the beginning of a sentence so that the sentence does not start with a mathematical symbol, as is often accepted as bad style. I think this does not change the meaning of the sentence |
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:<math>\mathcal{P}(\mathbb{N})=\{\varnothing,\{1, 2\}, \{1, 2, 3\}, \{4\}, \{1, 5\}, \{3, 4, 6\}, \{2, 4, 6,\dots\},\dots\}.</math>
Indeed, <math>\mathcal{P}(\mathbb{N})</math> contains infinite subsets of <math>\mathbb{N}</math>, e.g. the set of all positive even numbers <math>\{2, 4, 6,\ldots\}=\{2k:k\in \mathbb{N}\}</math>, along with the [[empty set]] <math>\varnothing</math>.
Now that we have an idea of what the elements of <math>\mathcal{P}(\mathbb{N})</math> are, let us attempt to pair off each [[element (math)|element]] of <math>\mathbb{N}</math> with each element of <math>\mathcal{P}(\mathbb{N})</math> to show that these infinite sets are equinumerous. In other words, we will attempt to pair off each element of <math>\mathbb{N}</math> with an element from the infinite set <math>\mathcal{P}(\mathbb{N})</math>, so that no element from either infinite set remains unpaired. Such an attempt to pair elements would look like this:
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