Spatial Geometric Operations
A Venn diagram in set theory is used to intuitively represent relationships between sets, such as intersection, union, and complement. The diagram contains 9 subfigures arranged in a 3 x 3 grid. Each subfigure contains two sets, A and B, as well as the universal set U and complements (′). The green area represents the selected result of the set operation.
In GIS, analysis tools, especially overlay analysis tools, commonly use the following operation types.
The table below shows the correspondence between set-theoretic expressions and GIS semantics.
| Figure | Mathematical Expression | Meaning | GIS Semantics |
|---|---|---|---|
| ① Intersection | A ∩ B | The part that belongs to both A and B | Intersection / Intersect |
| ② Union | A ∪ B | All areas that belong to A or B, or to both | Union |
| ③ Complement | A′ | Not A | Erase (A from U) |
| ④ A minus B | A ∩ B′ | The part that belongs to A but not to B; equivalent to A − B | Difference / Erase |
| ⑤ B minus A | A′ ∩ B | The part that belongs to B but not to A; equivalent to B − A | Difference |
| ⑥ A or not B | A ∪ B′ | The area that belongs to A or does not belong to B | |
| ⑦ Not A or B | A′ ∪ B | The area that does not belong to A or belongs to B | |
| ⑧ Complement of intersection | (A ∩ B)′ | All areas that do not belong to both A and B at the same time; A′ ∪ B′ = (A ∩ B)′ | |
| ⑨ Complement of union | (A ∪ B)′ | The area that belongs to neither A nor B; A′ ∩ B′ = (A ∪ B)′ |