Set Theory Basics
Fundamental set operations and relations.
The core vocabulary and notation of set theory: union, intersection, complement, subsets, power sets and Cartesian products, with set difference, symmetric difference, cardinality, membership, disjoint sets and De Morgan's laws.
Use it when
- You need the notation and meaning of a set operation.
- You are translating between set language and logic or SQL.
Watch out for
- Complement is relative to a stated universal set; without one it is undefined.
- Notations differ between textbooks (A^c, A', and a bar over A all denote complement).
The data
14 rows
| Concept | Notation | Description |
|---|---|---|
| Union | A ∪ B | Elements in A or B or both. |
| Intersection | A ∩ B | Elements common to A and B. |
| Complement | A' | Elements not in A within the universal set. |
| Subset | A ⊆ B | Every element of A is also in B. |
| Superset | A ⊇ B | A contains all elements of B. |
| Power Set | P(A) | Set of all subsets of A. |
| Cartesian Product | A × B | Set of all ordered pairs from A and B. |
| Set Difference | A ∖ B | Elements in A that are not in B. |
| Symmetric Difference | A Δ B | Elements in exactly one of A or B. |
| Empty Set | ∅ | The set with no elements; it is a subset of every set. |
| Cardinality | |A| | The number of elements in A. |
| Element Of | x ∈ A | x is a member of A. |
| Disjoint Sets | A ∩ B = ∅ | Sets with no elements in common. |
| De Morgan's Laws | (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ | The complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements. |
Fields
The shape of this dataset, read from the data itself: every column, its type and a real value. Agents can use this to plan a query without fetching the whole document.
| Section | Field | Type | Example |
|---|---|---|---|
| data | concept | string | Union |
| data | notation | string | A ∪ B |
| data | description | string | Elements in A or B or both. |
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