By Herbert Amann

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**Example text**

5, some m0 := min(B). If m0 = 0, then min(A) = α(0) ≥ n0 ≥ min(A) , and hence n0 = α(0). So we can suppose that n0 > min(A) and so m0 ∈ N× . But then α(m0 − 1) < n0 ≤ α(m0 ) and, by the deﬁnition of α, we have α(m0 ) = n0 . 8 Proposition A countable union of countable sets is countable. Proof For each n ∈ N, let Xn be a countable set. 7, we can assume that the Xn are countably inﬁnite and pairwise disjoint. Thus we have Xn = {xn,k ; k ∈ N } with xn,k = xn,j for k = j, that is, xn,k is the image of k ∈ N ∞ under a bijection from N to Xn .

A nonempty subset A of X is closed under the operation , if A A ⊆ A, that is, if the image of A × A under the function is contained in A. 8 Examples (a) Let X be a set. Then composition ◦ of functions is an operation on Funct(X, X). (b) ∪ and ∩ are operations on P(X). 2) and is commutative if x y = y x for x, y ∈ X. 2) are unnecessary and we write simply x y z. 3, composition is an associative operation on Funct(X, X). It may not be commutative (see Exercise 3). (b) ∪ and ∩ are associative and commutative on P(X).

7 Let R be a relation on X and S a relation on Y . Deﬁne a relation R × S on X × Y by (x, y)(R × S)(u, v) :⇐ ⇒ (xRu) ∧ (ySv) for (x, y), (u, v) ∈ X × Y . Prove that, if R and S are equivalence relations, then so is R × S. 8 Show by example that the partially ordered set P(X), ⊆ may not be totally ordered. 9 Let A be a nonempty subset of P(X). 6(a)). 5 The Natural Numbers 5 29 The Natural Numbers In 1888, R. ) [Ded95] about the set theoretical foundation of the natural number system. It is a milestone in the development of this subject, and indeed one of the high points of the history of mathematics.