The problem asks to find a set that is:

1. A proper superset of set $A = \{2, 3, 4\}$, meaning it must contain all elements of $A$ plus at least one additional element.
2. A subset of set $B = \{1, 2, 3, 4, 5\}$, meaning it can only contain elements from $B$.

Solution Steps:

• Set $A = \{2, 3, 4\}$ is given.
• Set $B = \{1, 2, 3, 4, 5\}$ is also given.

To meet both conditions:

1. The new set must include all elements of $A$ (i.e., $2, 3, 4$).
2. It must include at least one more element from $B$ that is not already in $A$. The possible additional elements from $B$ are $1$ and $5$.

Possible Solution:

• One valid set would be $\{1, 2, 3, 4\}$. This is a proper superset of $A$ (since it contains all elements of $A$ and one additional element, $1$), and it is a subset of $B$.
• Another valid set could be $\{2, 3, 4, 5\}$, which is also a proper superset of $A$ and a subset of $B$.

Thus, two valid answers are:

• $\{1, 2, 3, 4\}$
• $\{2, 3, 4, 5\}$

Would you like further clarification or more details on how proper supersets and subsets work?

Here are 5 related questions you might find helpful:

1. What is the difference between a subset and a proper subset?
2. Can a set be a proper subset and a superset at the same time?
3. What is the power set of a given set?
4. How many proper subsets can be formed from a set with $n$ elements?
5. How can we determine the union and intersection of two sets?

Tip: When finding proper supersets, always check if there are extra elements that belong to the larger set but are not in the smaller set.