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Set Theory: Functions What are functions in set theory? “Consider the function f(x) = x3, i.e., f assigns to each real number its cube.” “Let g assign to each country in the world its capital city.” (Lipschutz, 1998, pp. 94-116). These are examples. There are three basic features of functions. The first feature is that one element, ai, of the set A is assigned to, aligned with one unique element, bi, of the set B. This is one-to-one, is injective – as when one value of x is matched with one value of x3; ai = x = 3 is matched with bi = f(x) = x3 = 27. For the countries, g (France) = Paris, g (Denmark) = Copenhagen. The second feature is that every element bi of B is to be aligned with some element ai of A. This is not one-to-one; this is onto, surjective. Examples illustrate this feature. Thus f(x) = x2 yields for ai = -x = -2 and ai = x = 2 the same number, bi = 4. For the countries, g (England) and g (Scotland) = London. There is convergence on 4 and on London. The third feature is that when a function is both one-to-one and onto, like f(x) = x3 and g (France) = Paris the function is a one-to-one correspondence, is bijective. If a function is a one-to-one correspondence, then it is invertible (reversible): f: A  B
and f-1: BA,
wherein all elements of the sets A and B are considered.The application of functions to adaptation takes two forms. The first is the narrow form: the function of adaptedness assigns to each entity an environmental aspect – or the function of adaptedness binds each entity, each particular, to a facet of the environment, another particular. In spite of the order difference, the basic structure requirement is evident: one entity, a biological particular, is separated from its essential property of adaptedness, which is separated from an environmental particular. Conifers and the Third Principle of Adaptation The third principle of adaptation is (nominalistically) that if entity x is adapted to a second entity y, then the second is adapted to the first – and to be logically valid, if the second, y, is adapted to the first, x, then the first is adapted to the second (so the two are adapted to each other). Now the issue is two-fold. Data will be presented to uphold the third principle. And the presentation will be recast into the mold of functions. In Figure 1 are presented three non-overlapping species of coastally restricted forests of North America (Laderman 1998). In the coastal belt between southern Alaska and Washington state there is the Alaska yellow-cedar (Charmaecyparis nootkatensis). Though abundant and in places occurring in pure stands, it has declined. This decline started 100 years ago but is slight now. South of this species the coast redwood (Sequoia sempervirens) is abundant in the fog-shrouded coastal region from Oregon well into California. In spite of logging its regrowth shows it to be well suited to its present locale. Before the Pleistocene glaciation it was widespread in the West (Axelrod, 1976). Though it failed to regain its widespread distribution after the ice melted, still it seems well established in its present locale. The distribution of the Atlantic white cedar (Chamaecyparis thyoides) is intermittent along the East coast and Gulf coast. But it is a successful species, its net growth amply exceeding loss due to logging. These non-overlapping species are adapted to their locales which are adapted to them (third principle, nominalistically). These three species and three locales make up two sets of entities, of particulars. Species and locale are bound together by being adapted to each other. As previously mentioned there are two ways to express this binding: the function of adaptedness assigns to each species its locale; the function of adaptedness binds each species to its locale. In Figure 1 the three species are a1, a2, a3 and the three locales are b1, b2, b3. The function of adaptedness binds a1 to b1, a2 to b2, a3 to b3 – i.e., f(a1) = b1, f(a2) = b2, f(a3) = b3. This is one-to-one and onto; so this is a one-to-one correspondence, is bijective. But the locales seem well delimited in a unique, non-overlapping, idiosyncratic manner to their species and support their species quite well. Thus the function of adaptedness is invertible; the function is reversible. This means that the function of adaptedness is a one-to-one correspondence both ways. In Figure 1 are shown the vast and almost identical distributions of the white spruce (Picea glauca) and the black spruce (Picea mariana) (Brockman and Merrilees, 2001). Here there is a set of two species and a set of just one locale. This is not one-to-one; this is onto, surjective – because for every element of B (there is only one) there is aligned some element of A (the two species). The elements of A converge, because f(a1) = b1 and f(a2) = b1 but a1  a2.
So the function of adaptedness does not bind these species to their
locale as strongly as for the previous three species because the function
is not invertible – the locale does not grip just one species.
The structure of adaptation differs distinctly between the first three
conifers and the spruces.
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