Given the problem, let's break it down step-by-step:

1. Understanding the Elements

We are given:

• Set $u = \{1, 2, 3, 4, 5, 6, 7\}$.
• $A = \mathcal{P}(u)$, where $\mathcal{P}(u)$ is the power set of $u$. So, $A$ consists of all the subsets of $u$.
• Relation $R$ is the subset relation on $A$ (denoted $\subseteq$), meaning one set is related to another if it's a subset.

We are asked to consider the set $B = \{\{1\}, \{2\}, \{2, 3\}\} \subseteq A$.

2. Understanding Upper Bounds

An upper bound of $B$ is any element $x \in A$ such that for all $b \in B$, $b \subseteq x$. In other words, every subset in $B$ must also be a subset of $x$.

We are asked to find the upper bounds of $B$ that contain exactly 4 elements.

3. Determining the Upper Bounds

The key is to determine all sets $x \in A$ such that $\{1\}, \{2\}, \{2, 3\} \subseteq x$, and $x$ contains exactly 4 elements.

• To satisfy the subset relation $\{1\} \subseteq x$, the set $x$ must include the element 1.
• To satisfy $\{2\} \subseteq x$, the set $x$ must include the element 2.
• To satisfy $\{2, 3\} \subseteq x$, the set $x$ must also include the element 3.

Thus, any upper bound $x$ must include the elements $1, 2, 3$.

4. Finding the 4-Element Sets

Since we need the set $x$ to contain exactly 4 elements, and we've already determined that it must contain the elements $1, 2, 3$, we are left with the choice of adding exactly one more element from the remaining elements in $u$ (which are $\{4, 5, 6, 7\}$).

So, the possible upper bounds of $B$ are:

• $\{1, 2, 3, 4\}$
• $\{1, 2, 3, 5\}$
• $\{1, 2, 3, 6\}$
• $\{1, 2, 3, 7\}$

5. Conclusion

The upper bounds of $B$ that contain exactly 4 elements are: $\{1, 2, 3, 4\}, \{1, 2, 3, 5\}, \{1, 2, 3, 6\}, \{1, 2, 3, 7\}$

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Would you like more details on any of the steps? Here are 5 related questions you could explore next:

1. What is the cardinality of the power set of a given set?
2. How do you define upper and lower bounds in poset theory?
3. Can you have multiple upper bounds for a given set?
4. What is the lattice structure of the power set of a set?
5. How do you determine the least upper bound (supremum) of a set?

Tip: The power set of a set with $n$ elements has $2^n$ elements, and it forms a lattice under the subset relation!