# 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 *a* ∈ *A*.*B* then *A* is called a subset of *B*: *A* ⊂ *B*.*improper subset* of itself. If it is important to distinguish between proper and improper subsets then in addition to *B* ⊂ *A**B* ⊆ *A*.*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*x* and, therefore, the implication *x* ∈ Ø ⇒ *x* ∈ *A**A*.

There are various operations that defined over sets: intersection *A*∩*B*, union *A*∪*B*, 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:

*x* ∈ *A ^{c}* iff,

*x*∈

*X*and

*x*∉

*A*.

Complements satisfy de Morgan's Laws:

(*A*∩*B*)^{c} = *A ^{c}*∪

*B*and (

^{c}*A*∪

*B*)

^{c}=

*A*∩

^{c}*B*.

^{c}Sets may be finite or infinite.

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

|Contact| |Front page| |Contents| |Up|

Copyright © 1996-2018 Alexander Bogomolny64255400 |