- Appendix B: Set Theory for G-and E-Space
- Princeton University Press
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APPENDIX B Set Theory for G- and E-Space In this appendix, we present some set theoretical operations that are mathematically valid for subsets of G-space and E-space. 1. E ⫽ e1 ⫻ e2 ⫻ ... em, the environmental space, is the Cartesian product of all the sets of possible values of the m environmental variables. 2. G is the set of all cells (defined by their coordinates) existing in the selected region of the world. 3. The set η(G) ⫽ N ⊆ E is the available niche space, i.e., the subset of E that actually exists in the region of the world under consideration (Jackson and Overpeck 2000). 4. The difference between the sets A and B is defined as A – B ⫽ A ∩ BC The following is a list of valid operations in E-space. η(G1 ∪ G2) ⫽ η(G1) ∪ η(G2) The environment of a union of areas is the union of the environments of each separate area. η(G1 ∩ G2) ⊆ η(G1) ∩ η(G2) The environment of an intersection of areas is contained in the intersection of the environments of the separate areas. η(G1 – G2) ⊇ η(G1) – η(G2) ⫽ The environment of the difference of two η(G1) ∩ η(G2)C areas is contained in the intersection of the environment of the first with the complement of the environment of the second. η(G1 C) ⊇ N – η(G1) ⫽ The environment of the complement of an N ∩ η(G1)C area is contained in the intersection of the niche space with the complement of the environment of the area. G1 ⊆ G2 ⇒ η(G1) ⊆ η(G2) If an area is contained in another, its environment is contained in the environment of the other. SET THEORY FOR G- AND E-SPACES 267 If no repeated elements are present in E, then the preceding symbols for containing become equalities. The following is a list of valid G-space operations. η–1(N) ⫽ G The area corresponding to the entire niche space is the region in consideration. η–1(E1 ∪ E2) ⫽ η–1(E1) ∪ η–1(E2) The area corresponding to the union of two environmental sets is equal to the union of the areas of the two environmental sets, taken separately. η–1(E1 ∩ E2) ⫽ η–1(E1) ∩ η–1(E2) The area corresponding to the intersection of two environmental sets equals the intersection of the areas of the two environmental sets, taken separately. η–1(N – E1) ⫽ G – η–1(E1) The area corresponding to the complement (with respect to the available niche space) equals the geographic region minus the area corresponding to the environmental subset in question. These operations are mathematically valid. However, their biological interpretation contains several subtleties, as is discussed in the corresponding chapters. In addition, we stress that in general although both the operations for obtaining the environments of sets of geographic cells η(G′) ⫽ E′ and the geographic cells corresponding to sets of environmental vectors η–1(E′) ⫽ G′ can be implemented in a GIS, and they are in a sense inverse operations, one cannot simply assume that η–1[η(G′)] ⫽ η–1(E′) ⫽ G′. For this equality to be valid, a one-to-one correspondence must exist between environment vectors and geographic cells. If different geographic cells present the same environmental conditions , then the equality may not be valid. ...

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