What is a subset

Subset

 

The mathematical terms Subset and Superset describe a relationship between two sets. Another word for subset is Subset.

For the mathematical mapping of the embedding of a subset in its basic set, the mathematical function of the Subset relationship, the inclusion map is used. A. is a subset of B. and B. is a superset of A.if each element of A. also in B. is included. If B. also contains other elements that are not in A. are included so is A. a real subset of B. and B. is a real superset of A.. The amount all Subsets of a given set A. is called the power set of A..

The term Subset coined Georg Cantor - the 'inventor' of set theory - from 1884; the symbol of the subset relation was introduced by Ernst Schröder in 1890 in his "Algebra of Logic".

Notations and ways of speaking

(A. is a subset of B.), is a variant of the symbol
(A. is a real subset of B.)
(B. is a superset of A.)
(B. is proper superset of A.)

This notation emphasizes the analogy to the spellings xy and x < y. Often, however, no distinction is made between subset and real subset in terms of notation and the symbol ⊂ is then used for (any) subset. This use can be expanded as follows, so that, if necessary, you can again distinguish between subset and real subset using notation:

stands for "is a subset of",
stands for "is a real subset of". Variants of the symbol are .

The negation belonging to the other convention is of the latter symbol („ is no Subset of “) To distinguish.

In the situation one also often says:

A. is in B. contain "or"A. is from B. includes "or"B. contains A."Or"B. includes A." or similar.

For one also says accordingly:

A. is really in B. contain "or"A. is from B. real includes "or"B. contains A. really "B. includes A. real".

When using such ways of speaking, it must be ensured that in connection with the element relation Sometimes the same or similar ways of speaking are used, which can lead to confusion.

definition

This means: "A is by definition a subset of B if and only if every element of A is also an element of B."
This means: “A is by definition a real subset of B if and only if every element of A is also an element of B and at the same time at least one element of B Not Element of A. "

Examples

The set {drum, playing card} is a subset of the set {guitar, playing card, digital camera, drum}
The regular polygons are a subset of the set of all polygons.
  • {1, 2} is one (real) Subset of {1, 2, 3}.
  • {1, 2, 3} is one (fake) Subset of {1, 2, 3}.
  • {1, 2, 3, 4} is not a subset of {1, 2, 3}.
  • {1, 2, 3} is not a subset of {2, 3, 4}.
  • {} is a (real) Subset of {1, 2}.
  • {1, 2, 3} is one (real) Superset of {1, 2}.
  • {1, 2} is one (fake) Superset of {1, 2}.
  • {1} is not a superset of {1, 2}.
  • The set of prime numbers is a real subset of the set of natural numbers.
  • The set of rational numbers is a proper subset of the set of real numbers.

Further examples as quantity diagrams:

  • A is a proper subset of B

  • C is a subset of B, but not a real subset of B

properties

Inclusion as a relation of order

If A ⊆ B and B ⊆ C, then A ⊆ C too

The inclusion as a relation between sets fulfills the three properties of a partial order relation, namely it is reflexive, antisymmetric and transitive:

(It is an abbreviation for " and “.)

Is so a set of sets (a set system) then is a partial order. This is especially true for the power set a given amount .

Chains of inclusion

Is a system of quantities, so that every two of the in Occurring quantities which one includes the other or is comprised by the other, one calls such a system of quantities one Inclusion chain. The system provides an example of this the left-hand unrestricted open intervals of .

A special case of an inclusion chain is when a (finite or infinite) sequence of sets is given which is capable of ascending or fortune descending is arranged. You then write briefly:

Size and number of subsets

See also

  • characteristic function
Based on an article in: Wikipedia.de
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Date of the last change: Jena, the: 21.10. 2019