részhalmazreláció

The subset relation, denoted by $\subseteq$, is a fundamental concept in set theory. For any two sets A and B, A is considered a subset of B if and only if every element of A is also an element of B. This can be formally expressed as: $A \subseteq B \iff \forall x (x \in A \implies x \in B)$.

If set A contains at least one element that is not in set B, then A is not a subset of B, which is denoted by $A \not\subseteq B$. Every set is a subset of itself, meaning $A \subseteq A$ is always true. The empty set, denoted by $\emptyset$ or {}, is a subset of every set. This is because the condition "every element of the empty set is also an element of set B" is vacuously true, as there are no elements in the empty set to contradict the implication.

The subset relation is reflexive ($A \subseteq A$), antisymmetric (if $A \subseteq B$ and $B \subseteq A$, then $A = B$), and transitive (if $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$). These properties mean that the subset relation forms a partial order on the power set of a given set.

A proper subset is a subset that is not equal to the original set. If $A \subseteq B$ and $A \neq B$, then A is a proper subset of B, denoted by $A \subset B$. This implies that B must contain at least one element not present in A.