The problem in the image states:

Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ and $A = \{1\}$. Find the set $A^c$.

Solution:

• $U$ is the universal set, and it contains all the elements: $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.
• $A$ is a subset of $U$, specifically $A = \{1\}$.
• $A^c$, the complement of $A$, consists of all the elements in $U$ that are not in $A$.

Thus, to find $A^c$, we subtract the elements of $A$ from $U$: $A^c = U - A = \{2, 3, 4, 5, 6, 7, 8, 9\}.$

Final Answer:

$A^c = \{2, 3, 4, 5, 6, 7, 8, 9\}.$

Would you like further details or clarification?

5 Related Questions:

1. What is the complement of a set in general set theory?
2. How do you find the union of two sets?
3. What is the intersection of sets $A$ and $B$ if $A = \{1\}$ and $B = \{3, 4\}$?
4. Can the complement of a set ever be larger than the universal set?
5. How would you represent the complement of a set visually using a Venn diagram?

Tip:

Always check that the union of a set and its complement equals the universal set. This ensures your complement is calculated correctly!