b) Use the idea in part (a) to determine an explicit one-to-one correspondence between zt and the set of all ordered pairs of positive integers.
b) Use the idea in part (a) to determine an explicit one-to-one correspondence between zt and the set of all ordered pairs of positive integers.
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**b)** Use the idea in part (a) to determine an explicit one-to-one correspondence between ℤ⁺ and the set of all ordered pairs of positive integers.
**c)** Use part (a) to give a different proof of Theorem 1.26.
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Transcribed Image Text:**Educational Text Transcription:**
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14. Interpret the set of all ordered pairs of positive integers as a grid of dots in the first quadrant of the xy-plane. Consider the "path" that traverses these dots in the following order:
(1, 1), (2, 1), (1, 2), (3, 1), (2, 2), (1, 3), (4, 1), (3, 2), (2, 3), (1, 4), (5, 1), …
a) Show that this constitutes a proof that the set of all ordered pairs of positive integers is countably infinite.
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**Explanation:**
The list of ordered pairs (x, y) begins at (1, 1) and moves along diagonals in the grid. This sequence demonstrates how every possible pair of positive integers can be systematically listed, showing that they correspond to the natural numbers through this process. This is a classic method for proving the countability of the set of all ordered pairs of positive integers.
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