Subset

The mathematical concepts superset and subset describe a relationship between two quantities. Another word for subset is a subset.

For the mathematical representation of the embedding of a subset in their basic amount, the mathematical function of the subset relation, the inclusion mapping is used. A is a subset of B and B is a superset of A, when each element of A is included in B. If B also contains other elements that are not contained in A, then A is a proper subset of B and B is a proper subset of A. The set of all subsets of a given set A is called the power set of A.

The term coined subset 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

This notation emphasizes the analogy with the notation x ≤ y and x

From the latter the symbol belonging to another convention negation ( " is not a subset of" ) is to be distinguished.

In the situation we also say often:

For they say accordingly:

Using these ways of speaking, make sure that sometimes the same or similar ways of speaking are used in conjunction with the element - relation, which can possibly lead to confusion.

The corresponding Unicode symbols are: ⊂, ⊃, ⊆, ⊇ (see: Unicode block Mathematical operators ).

Definition

Examples

• {1, 2 } is a ( real ) subset of {1, 2, 3}.
• {1, 2, 3 } is a (non-genuine ) 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 a ( real ) superset of {1, 2}.
• {1, 2 } is a (non-genuine ) superset of {1, 2}.
• {1 } is not a superset of {1, 2}.

• The set of prime numbers is a proper subset of the set of natural numbers.
• The set of rational numbers is a proper subset of the set of real numbers.

Properties

• The empty set is a subset of any amount:
• Any amount is a subset of itself:
• Characterization of inclusion with the help of the Association:
• Characterization of inclusion with the help of the average:
• Characterization of inclusion using the changeset:
• Characterization of inclusion by means of the characteristic function:
• Two sets are equal if and only if each is a subset of the other: This rule is often used in the detection of the equality of two quantities by showing the mutual inclusion ( in two steps ).
• On transition to complement the direction of inclusion is reversed:
• In the formation of intersection always obtained a subset:
• In the formation of the union always gives a superset:

The inclusion as order relation

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

(This is a shorthand for "and". )

So is a set of sets ( a set system ), then a partial order. This is particularly true for the power set of a given set.

Inclusion chains

Is an amount of the system, so that of two of the quantities occurring in the one comprises the other or is surrounded by the other, it is called an amount of such an inclusion system chain. An example of this is the system of left-sided unbounded open intervals.

A special case of an inclusion chain is when a (finite or infinite ) sequence of sets is given, which by virtue of ascending or virtue is arranged in descending order. It then writes briefly:

Size and number of subsets

• Every subset of a finite set is finite and is valid for the widths:
• Any superset of an infinite set is infinite.
• Even with infinite sets is valid for the widths:
• For infinite sets but it is possible that a proper subset of the same cardinality as their base set. For example, the natural numbers are a proper subset of the integers, but the two sets are equally powerful ( ie countably infinite).
• The power set of a set A is always more powerful than the set A itself.
• A finite set with n elements has exactly 2n subsets.
• The number of k- element subsets of an n-element (finite) quantity is given by the binomial coefficients.