 Posted by By Herbert Amann

ISBN-10: 3764371536

ISBN-13: 9783764371531

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If X is also a partially ordered set, then f is called bounded on bounded sets if, for each bounded subset A of X, the restriction f |A is bounded. 7 Examples (a) Let X and Y be sets and f ∈ Y X . 8 says that the induced functions f : P(X) → P(Y ) and f −1 : P(Y ) → P(X) are increasing. (b) Let X be a set with at least two elements and X := P(X)\{X} with the inclusion order. Then the identity function X → X , A → A is bounded on bounded sets but not bounded. Operations A function : X × X → X is often called an operation on X.

G(x) , It is clear that is associative or commutative whenever the same is true of If Y has an identity element e with respect to , then the constant function X→Y , . x→e is the identity element of Funct(X, Y ) with respect to . Henceforth we will use the same symbol for the operation on Y and for the induced operation on Funct(X, Y ). From the context it will be clear which function the symbol represents. We will soon see that this simple and natural construction is extremely useful. 14(b). Exercises 1 Let ∼ and ∼ ˙ be equivalence relations on the sets X and Y respectively.

This implies that a ≥ n + 1 for all a ∈ A, that is, n + 1 ∈ B. Because of the induction axiom (N1 ) we have B = N. But this implies that A = ∅ because, if m ∈ A, then m ∈ N = B which means that m is a lower bound and, hence a minimum element, of A, which is not possible. We have therefore found the desired contradiction: A = ∅ and A = ∅. For an example of the use of the well ordering principle, we discuss the prime factorization of natural numbers. We say that a natural number p ∈ N is prime if p ≥ 2 and p has no divisors except 1 and p.