To find $(A \cap C)'$ using the roster method, we will follow these steps:

Step 1: Find the intersection $A \cap C$

The intersection of sets $A$ and $C$, denoted as $A \cap C$, contains all the elements that are present in both sets.

Given:

• $A = \{1, 3, 4, 8\}$
• $C = \{1, 2, 4, 5, 8\}$

Now, let's find $A \cap C$:

• Elements common in both $A$ and $C$ are: $\{1, 4, 8\}$

So, $A \cap C = \{1, 4, 8\}$.

Step 2: Find the complement of $A \cap C$, denoted as $(A \cap C)'$

The complement of a set $X$ (denoted as $X'$) consists of all the elements in the universal set $U$ that are not in $X$.

Given:

• Universal set $U = \{1, 2, 3, 4, 5, 6, 7, 8\}$
• $A \cap C = \{1, 4, 8\}$

Now, find $(A \cap C)'$:

$(A \cap C)' = U - (A \cap C) = \{2, 3, 5, 6, 7\}$.

Answer:

$(A \cap C)' = \{2, 3, 5, 6, 7\}$

Would you like to know more details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How do you find the union of two sets using the roster method?
2. What is the difference between the intersection and the union of sets?
3. How do you find the complement of a set when given in a different context?
4. Can you find the symmetric difference of sets $A$ and $C$?
5. How would the answer change if the universal set $U$ had different elements?

Tip: When finding a complement of a set, always remember to compare it against the universal set to ensure accuracy.