1. Let's start by stating the problem: We want to understand the difference between a proper set and a subset. 2. A subset is a set where every element of set A is also in set B. We write this as $A \subseteq B$ which means A is a subset of B. This includes the possibility that A and B are exactly the same. 3. A proper subset is a subset that is strictly smaller than the other set. This means every element of A is in B, but B has at least one element not in A. We write this as $A \subset B$. 4. Important rule: If $A \subseteq B$ and $A \neq B$, then $A$ is a proper subset of $B$. 5. In simple terms, a subset can be equal to the set itself, but a proper subset cannot be equal; it must be smaller. 6. Example: If $B = \{1,2,3\}$, then $A = \{1,2\}$ is a proper subset of $B$ because $A \subset B$. 7. But $B$ is also a subset of itself because $B \subseteq B$. Final answer: A subset can be equal to the set, but a proper subset must be strictly smaller than the set.