Figure 1
The goals of intra-site analysis based on the relationships between the basic elements of a site.
Table 1
Definition and notation for common set properties and operations.
| Operation | Symbol | Example | Definition |
|---|---|---|---|
| Set | {…} | a, b, c, d | A collection of elements |
| Cardinality | |…| | |a, b, c| = 3 | # of elements in a set |
| Empty set | ∅ | {}, |∅|=0 | A set with no elements |
| Universe | Ω or u | Ω({a, b, c}) = {a, b, c, d, …, z} | Set of all possible elements in a set’s domain |
| Union | ∪ | {a, b, c} ∪ {c, d} = {a, b, c, d} | Elements in either A or B |
| Intersection | ∩ | {a, b, c} ∩ {c, d} = {c} | Elements in both A and B |
| Symmetric Difference | Δ | {a, b, c} Δ {c, d, e} = {a, b, d, e} | Elements in either A or B, but not in both |
| Multiset Sum | ⊎ | {a1, b2} ⊎ {a2, b1} = {a3, b3} | Sum of the multiplicities of multisets |
| Subset, Superset | ⊆, ⊇ | {a, b} ⊆ {a, b, c} ⊆ {a, b, c} | A contains some, or all, elements in B |
| Strict Sub-, Superset | ⊂, ⊃ | {a, b} ⊂ {a, b, c} ⊄ {a, b, c} | A contains some, but not all, elements in B |
| “in…” | ∈ | a ∈ {a, b, c} | a is an element in a set |
| “for all…” | ∀ | ∀ x ∈ ℕ = {0, 1, 2, 3, …, ∞} | Logical statement used in set declarations |
| “and” | ∧ | ∀ x ∈ ℕ ∧ x ≤ 2 = {0, 1, 2} | Logical statement used in set declarations |
| “such that…” | | or : | ∀ x ∈ ℕ |0 < x ≤ 2 = {1, 2} | Logical statement used in set declarations |
| Complement | A′ or Ac | A′ = {∀a | a ∉ A} | All elements not in A |
| Powerset | ℘(A) | ℘({1, 2}) = {∅, {1}, {2}, {1, 2}} | All possible subsets of a set |
Figure 2
A Venn diagram of unions and intersections for two sets, A and B and their complements, within a universe ΩA,B.
Figure 3
Difference between set union ∪, multiset union ∪, and multiset sum ⊎. Note the handling of intersecting elements (element C).
Figure 4
Example hypergraph H with seven vertices/nodes V(H) = {v1, …, v7} and four edges E(H) = {e1, …, e4}, where each edge ei connects a subset of vertices vi (User:Kilom691 2010).