Question

the symbols a ⊂ b mean select an answer while the symbols a ⊆ b mean select an answer. both proper and improper subsets are subsets according to the definition that you found above, but vary in an important way. if we call a an improper subset of b, then a select an answer. on the other hand, if a is a proper subset of b, then a select an answer. if we use the symbols a ⊂ b that means that the subset select an answer this symbol can be used with select an answer. if we use the symbols a ⊆ b that means that the subset select an answer this symbol can be used with select an answer.

Explanation:

The symbol $A\subset B$ means that $A$ is a proper subset of $B$, i.e., all elements of $A$ are in $B$ and $B$ has at least one element not in $A$. The symbol $A\subseteq B$ means that $A$ is a subset of $B$, which includes the case where $A = B$ (improper subset) and proper - subset cases. If $A$ is an improper subset of $B$, then $A=B$. If $A$ is a proper subset of $B$, then $A
eq B$. The symbol $A\subset B$ is used only for proper subsets. The symbol $A\subseteq B$ can be used for both proper and improper subsets.

Answer:

1. $A$ is a proper subset of $B$
2. $A$ is a subset of $B$
3. $=B$
4. $

eq B$

1. is proper
2. only proper subsets
3. can be proper or improper
4. both proper and improper subsets