Exercise 1.2.4

Question

Produce an infinite collection of sets $A_{1}, A_{2}, A_{3}, \ldots$ with the property that every $A_{i}$ has an infinite number of elements, $A_{i} \cap A_{j}=\emptyset$ for all $i 
eq j$, and $\bigcup_{i=1}^{\infty} A_{i}=\mathbf{N}$

Ulisse Mini · Accepted Answer

This question is asking us to partition $\mathbf N$ into an infinite collection of sets.
  This is equivalent to asking us to unroll $\mathbf N$ into a square, which we can do along the diagonal
  $$
  \begin{array}{lccccc}
    1 & 3 & 6 & 10 & 15 & \cdots \
    2 & 5 & 9 & 14 & \cdots & \
    4 & 8 & 13 & \cdots & & \
    7 & 12 & \cdots & & & \
    11 & \ldots & & & & \
    \vdots & & & & &
  \end{array}
  $$

Exercise 1.2.4

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Exercise 1.2.4

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