To solve $(A \cap B) \cap C$, we will proceed step by step:

Step 1: Compute $A \cap B$

The intersection of two sets $A$ and $B$ is the set of elements common to both $A$ and $B$.

$A = \{0, 3, 5\}, \, B = \{5, 7\}$ Common element: $\{5\}$.

Thus: $A \cap B = \{5\}.$

Step 2: Compute $(A \cap B) \cap C$

Now, we take the intersection of the result $\{5\}$ from Step 1 with $C = \{0, 7\}$. The intersection is the set of elements common to both.

$(A \cap B) = \{5\}, \, C = \{0, 7\}$ There are no common elements between these sets.

Thus: $(A \cap B) \cap C = \emptyset$

Final Answer:

$(A \cap B) \cap C = \emptyset$

Would you like a detailed explanation or examples of similar problems?

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Relative Questions:

1. What is the difference between union and intersection in set theory?
2. How do we calculate the complement of a set?
3. Can you solve $(A \cup B) \cap C$ for the same sets $A$, $B$, and $C$?
4. How does the distributive property work in set operations?
5. Can intersections and unions be visualized using Venn diagrams?

Tip:

To verify set operations, listing all elements in each step clearly helps avoid errors!