2.12 ADDITIONAL EXERCISES
- Prove (without looking up Euclid’s proof) that there are infinitely many primes. Hint. See Example 2.1.2.40 and Exercise 2.1.2.42.
- Modify the pairing function of Grzegorczyk (1953) (2.1.4.5) to make it onto.
- Give a direct proof that the J in 2.1.4.6 is 1-1.
- Give a direct proof that the J in 2.1.4.8 is 1-1.
- Show that J given by
is an onto pairing function in . Show that its projections are in .
Hint. View 2 is the union of all the finite groups of pairs, for i = 0, 1, 2, . . . ,
Gi = {〈x, y〉 ∈ 2 : x + y = i}
Now enumerate 2 by listing pairs 〈x, y〉, first, by ascending order (with respect to i) of their group-number i, and then, within each group Gi, listing them in ascending order of the y-component. Show that the position n = 0, 1, 2, . . . of 〈x, y〉 in this enumeration is precisely J(x, y), which settles onto-ness. Of course, projections K and L exist (why?). ...
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