If A and B are stes, then `A cap (A cup B)=`

Answer

To solve the problem \( A \cap (A \cup B) \), we will follow these steps:

### Step 1: Understand the operations
We need to understand what the intersection (\( \cap \)) and union (\( \cup \)) operations mean in set theory:
- The union of two sets \( A \cup B \) includes all elements that are in either set \( A \), set \( B \), or in both.
- The intersection of two sets \( A \cap B \) includes only the elements that are in both sets.

### Step 2: Apply the definition of union
We start with the expression \( A \cap (A \cup B) \). According to the definition of union:
\[ A \cup B = \{ x | x \in A \text{ or } x \in B \} \]
This means that \( A \cup B \) contains all elements of set \( A \) and all elements of set \( B \).

### Step 3: Apply the definition of intersection
Now we need to find \( A \cap (A \cup B) \). This means we are looking for elements that are in set \( A \) and also in the set \( A \cup B \):
\[ A \cap (A \cup B) = \{ x | x \in A \text{ and } (x \in A \text{ or } x \in B) \} \]

### Step 4: Simplify the expression
Since any element \( x \) that is in set \( A \) will also be in the union \( A \cup B \), we can simplify the expression:
\[ A \cap (A \cup B) = A \]

### Conclusion
Thus, we conclude that:
\[ A \cap (A \cup B) = A \]