Set Theory
Set theory is the branch of mathematical logic that studies collections of objects, called sets, and the relationships between them. It provides the foundational language for nearly all of modern mathematics, defining concepts like numbers, functions, and relations in terms of sets. Developed formally by Georg Cantor in the 1870s, it underpins areas from algebra and topology to computer science and logic.
Common Set Operations and Their Notations
| Operation | Symbol | Meaning | Example (A={1,2}, B={2,3}) |
|---|---|---|---|
| Union | ∪ | All elements in A or B | A ∪ B = {1,2,3} |
| Intersection | ∩ | Elements in both A and B | A ∩ B = {2} |
| Difference | \ | Elements in A but not B | A \ B = {1} |
| Complement | Aᶜ | Elements not in A (within universal set) | If U={1,2,3,4}, Aᶜ={3,4} |
| Cartesian Product | × | All ordered pairs from A and B | A × B = {(1,2),(1,3),(2,2),(2,3)} |
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Related Terms
Subset
A subset is a set whose every element is also an element of another set, called the superset. If A is a subset of B (written A ⊆ B), then for every element x in A, x is also in B. The concept of subsets is fundamental to set theory and appears throughout mathematics in topics like topology, algebra, and probability theory.
Set Union
The union of two sets A and B is the set containing all elements that are in A, in B, or in both, with no element repeated. Written as A ∪ B, it combines the members of both sets into a single collection. Set union is used in probability (combining events), database queries (SQL OR), and logic circuit design.
Set Intersection
The intersection of two sets A and B is the set containing only those elements that are members of both A and B simultaneously. Written as A ∩ B, it identifies common elements shared between collections. Set intersection is used in database queries (SQL JOIN/AND), probability (joint events), and data deduplication tasks.
From German "Mengenlehre" (Menge = collection/quantity + Lehre = doctrine/theory), coined by Georg Cantor circa 1874. "Set" in English derives from Old English "settan" (to place together).