By Mary Ellen Rudin
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Extra info for Lectures on Set Theoretic Topology
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38. Let m and n E w and let there be a bijection f: n ~ m. Thenm = n. Proof Let w' ={new :mewandf:n~mabijection=> m =n}. Now, since the image by any map of the null set is the null set, a map f: 0 ~ m is bijective only if m = 0. So 0 E w'. Let nEw' and let f: n u {n} ~ m be a bijection. Since m ~ 0, m = k u {k}, for some k E w. Define f': n ~ k by f'(i) = f(i) if f(i) ~k, andf'(i) = f(n) if f(i) = k. Thenf' is bijective, withj'- 1(j) = f- 1(j) ifj ~ f(n) andj'- 1(j) = f- 1(k) ifj = f(n). Since nEw' it follows that k = n.
Let S be a non-null subset of Z such that, for any a, bE S, 51 FURTHER EXERCISES + a b and b - a E S. Prove that there exists d E w such that S = {nd: n E Z}. (Cf. 80. Let p be a prime number and let n be a positive number that is not a multiple of p. Prove that there exist h, k E Z such that hn +kp = 1. 81. Prove that Z21 is a field if, and only if, p is a prime. 82. Let p be a prime number. Prove that, for all a, b E w, if p divides ab, then p divides either a or b. 83. Prove that any field not of characteristic zero has prime characteristic.
The notation m" is here used in two senses, to denote both the set of maps n ~ m and the cardinality of this set. The latter usage is the original one. The use of the notation yx to denote the set of maps X~ Y is much more recent and was suggested by the above formula, by analogy with the use of X to denote cartesian product. The number m" is called the nth power of m. It may be defined recursively, for each m E ro, by the formula m0 = 1, m1 = m and, for all k E ro, mk+ 1 = (mk)m. For all m, n, pEw, (mn)t> = mvnv, mn+v = mnmP and mnP = (mn)P.