Given the universal set $U = \{ 4, 5, 6, 7, 8, 9, 10 \}$ and the subset $A = \{ 5, 6, 9, 10 \}$, let's explore the properties of set $A$ within the context of $U$.

Analysis of Set $A$:

1. Elements of Set $A$: The set $A$ consists of the elements 5, 6, 9, and 10.
2. Subset Relationship: Since all elements of $A$ are also in $U$, $A$ is a subset of $U$, denoted as $A \subseteq U$.
3. Complement of $A$: The complement of $A$ in $U$, denoted as $A'$ or $U \setminus A$, is the set of elements in $U$ that are not in $A$. So, $A' = \{ 4, 7, 8 \}$
4. Cardinality of $A$: The number of elements in $A$ is $|A| = 4$.
5. Cardinality of $U$: The number of elements in $U$ is $|U| = 7$.

Would you like further details on this analysis or have any specific questions?

Relative Questions:

1. What is the union of sets $A$ and $A'$?
2. How do you find the intersection of sets $A$ and $A'$?
3. What is the cardinality of the complement of set $A$?
4. Can you list all possible subsets of set $A$?
5. How would you represent set $A$ in terms of a Venn diagram?

Tip:

When dealing with sets, always ensure you clearly define the universal set $U$ as it provides the context for operations like complements and unions.