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 into its basic set, the mathematical function of the subset relation, the inclusion mapping is used. is a subset of and is a superset of if every element of also contained in Moreover, if contains other elements not contained in then is a true subset of and is a true superset of . The set of all subsets of a given set is called the power set of .

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

Set diagram: A is a (real) subset of B.

Definition

If and are sets and each element of also an element of , then called a subset or subset of :

$A \subseteq B :\Longleftrightarrow \forall x \in A\colon x \in B$

Conversely, called the superset of exactly when is subset of

$B \supseteq A :\Longleftrightarrow A \subseteq B$

Furthermore, there is the notion of a real subset. is a real subset of exactly if a subset of and is not identical to }.

$A\subsetneq B:\Longleftrightarrow A\subseteq B\land A\neq B$

Again, one also writes $B \supsetneq A$ if $A \subsetneq B$ .

Other notations

⊂⊊⊆⊇⊋⊃

Some authors also use the characters $\subset$ and $\supset$ for subset and superset instead of $\subseteq$ and $\supseteq$ . Mostly the author then does not define the term "real subset".

Other authors prefer the characters $\subset$ and $\supset$ for true subset and superset thus instead of $\subsetneq$ and $\supsetneq$ . This usage is fittingly reminiscent of the characters for inequality ≤ $\leq$ and . Since this notation is mostly used when the difference between real and non-real subsets is important, the characters $\subsetneq$ and $\supsetneq$ rather rarely used.

Variants of the character $\subsetneq$ are also $\varsubsetneq$ , $\subsetneqq$ and $\varsubsetneqq$ . If is not a subset of , can also be $A\nsubseteq B :\Longleftrightarrow \lnot \left(A\subseteq B\right)$ used. Corresponding notations are $\varsupsetneq$ for $\supsetneq$ , $\supsetneqq$ and $\varsupsetneqq$ for $\supsetneq$ , and $A\nsupseteq B$ (no superset).

The corresponding Unicode symbols are: ⊂, ⊃, ⊆, ⊇, ⊄, ⊅, ⊈, ⊉, ⊊, ⊋ (see: Unicode block Mathematical Operators).

Speech

Instead of " is a subset of ." we also say "The set contained in the set " or "The set is contained by ." said. Similarly, instead of saying " is a superset of .", we say "The set contains the set ." or "The set contains the set .". If there can be no misunderstanding, " contains ." etc. is also said. Misunderstandings can arise in particular with "The set contains the element .".

Examples

{1, 2} is a (real) subset of {1, 2, 3}.
{1, 2, 3} is a (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 a (real) superset of {1, 2}.
{1, 2} is a (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 real subset of the set of real numbers.

More examples as set diagrams:

A is a real subset of B

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

The set {drum, playing card} is a subset of the set {guitar, playing card, digital camera, drum}

The regular polygons form a subset of the set of all polygons.

Properties

The empty set is a subset of each set:

$\varnothing \subseteq A$

Every set is a subset of itself:

$A \subseteq A$

Characterization of inclusion with the help of the association:

$A \subseteq B \Leftrightarrow A \cup B = B$

Characterization of inclusion using the average:

$A \subseteq B \Leftrightarrow A \cap B = A$

Characterization of inclusion using the difference set:

$A \subseteq B \Leftrightarrow A \setminus B = \varnothing$

Characterization of inclusion using the characteristic function:

$A \subseteq B \Leftrightarrow \chi_A \le \chi_B$

Two sets are equal if and only if each is a subset of the other:

$A=B\Leftrightarrow A\subseteq B\land B\subseteq A$

This rule is often used when proving equality of two sets by showing mutual inclusion (in two steps).

In the transition to complement, the direction of inclusion reverses:

$A \subseteq B \Rightarrow A^{\rm c} \supseteq B^{\rm c}$

When forming the intersection, you always get a subset:

$A \cap B \subseteq A$

When forming the union set, you always get a superset:

$A \cup B \supseteq A$

Inclusion as order relation

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

$A \subseteq A$

$A \subseteq B \subseteq A \Rightarrow A = B$

$A \subseteq B \subseteq C \Rightarrow A \subseteq C$

(Where $A\subseteq B\subseteq C$ a shorthand notation for $A\subseteq B$ and $B\subseteq C$ .)

Thus, if $M \,$ a set of sets (a set system), then $(M, \subseteq)$ a half-order. In particular, this holds for the power set $\mathcal P(X)$ a given set .

If A ⊆ B and B ⊆ C, then also A ⊆ C

Inclusion Chains

If is $M \,$ a set system such that of every two sets occurring in $M \,$ one includes or is included by the other, such a set system is called an inclusion chain. An example of this is provided by the system $\{{]{-\infty, x}[} \mid x \in \R \}$ of the left unconstrained open intervals of $\mathbb {R}$ .

A special case of an inclusion chain exists if a (finite or infinite) set sequence is given which is ordered by $\subseteq$ ascending or by $\supseteq$ descending. One writes then briefly:

$A_1 \subseteq A_2 \subseteq A_3 \subseteq \ ...$

$A_1 \supseteq A_2 \supseteq A_3 \supseteq \ ...$

Size and number of subsets

Every subset of a finite set is finite and for the powers holds:

$A \subseteq B \Rightarrow \left| A\right| \le \left| B\right|$

$A \subsetneq B \Rightarrow \left| A\right| < \left| B\right|$

Every superset of an infinite set is infinite.
The same applies to the thicknesses for infinite sets: