What Is Set?

Set, is a basic concept of mathematics. The concept of a set is inseparable from a concept of an element. Sets have (or contain) elements, elements belong to sets. Roughly speaking, the terms set, collection, conglameration, class, assembly, group, pile, heap and such might have been interchangeable, except that some of them have acquired special meanings in mathematics.

The fact that element a belongs to set A is expressed as aA. If all elements of set A also belong to set B then A is called a subset of B: AB. Every set is a subset of itself: A ⊂ A. As such, it is called an improper subset of itself. If it is important to distinguish between proper and improper subsets then in addition to BA we sometimes use BA. If the latter is used then BA implies B ≠ A.

Algebraically, A ⊆ B is equivalent to either A = A∩B or B = A∪B.

The empty set - Ø - that has no elements is a subset of every set. This is because x ∈ Ø is false for any x and, therefore, the implication x ∈ Ø ⇒ xA is true for any set A.

There are various operations that defined over sets: intersection AB, union AB, symmetric difference A^B. It is common to restrict consideration only to the subsets of a particular "large" set, say X, in which case we also introduce a unary operation c - passing to a complement:

xAc iff, xX and xA.

Complements satisfy de Morgan's Laws:

(AB)c = AcBc and (AB)c = AcBc.

Sets may be finite or infinite.

The set of all subsets of set A is denoted by 2A. This is because the number of the subsets of a finite set A with n elements is exactly 2n.

Related material

Set Theory

  • Addition of Sets
  • Cantor-Bernstein-Schroeder theorem.
  • de Morgan's Laws
  • Equivalence Relations
  • Mutiplication of Sets
  • Nested Subsets
  • Russell's Paradox
  • Subsets and Intersections
  • The set of all subsets of a given set is bigger than the set itself
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