Subset

A part of a Set; Set A is a subset of B.

Definition

A set $A$ is a subset of $B$ if all elements of A are also elements of B.

$$A\subseteq B⟺\forall x(x\in A⟹x\in B)$$

Proper Subset

$$A\subset B\equiv A\subseteq B\wedge A\ne B$$