Mathematical logic and foundations Definition

Subset

Builds on Set Element Universal quantifier Implication

Definition

Let $A$ and $B$ be sets. The set $A$ is a subset of $B$ , written $A \subseteq B$ , when each element of $A$ is an element of $B$ : $\forall x \, (x \in A \to x \in B).$ Then $B$ is a superset of $A$ . The set $A$ is a proper subset of $B$ , written $A \subsetneq B$ , when $A \subseteq B$ and $A \neq B$ .

Remarks

Some authors write $A \subset B$ for $A \subseteq B$ , and others write it for a proper subset. Two sets are equal if and only if each is a subset of the other. This follows from the axiom of extensionality. The empty set is a subset of every set, and every set is a subset of itself.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Element definition
Proposition concept
Set definition
Implication definition
Predicate definition
Universal quantifier definition
Subset definition

Where this leads

1020 pages build on this one.

Tag	`0FE8Q5S42Y`
URI	`m2://0FE8Q5S42Y`
Kind	definition
Writing	W1 stated
MSC	03E20
Defines	Subset
Uses	Set, Element, Universal quantifier, Implication

Subset

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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