If A,B,C be three sets, then `A cap (B cup C) = ?`

Options

A. `(A cap B) cup (A cap C)`
B. `(A cap B) cup C`
C. `(A cap B) cup (A cap C)`
D. None of these

Answer

To solve the problem \( A \cap (B \cup C) \), we will follow a step-by-step approach.

### Step 1: Understand the expression
The expression \( A \cap (B \cup C) \) means we are looking for the elements that are common to set \( A \) and the union of sets \( B \) and \( C \).

### Step 2: Find \( B \cup C \)
The union of sets \( B \) and \( C \), denoted as \( B \cup C \), includes all elements that are in either set \( B \), set \( C \), or in both.

### Step 3: Find the intersection with \( A \)
Now, we need to find the intersection of set \( A \) with the result of \( B \cup C \). This means we are looking for elements that are in set \( A \) and also in the union of sets \( B \) and \( C \).

### Step 4: Visualize using a Venn Diagram
To visualize this, we can draw a Venn diagram with three circles representing sets \( A \), \( B \), and \( C \). The area that represents \( B \cup C \) will include all parts of circles \( B \) and \( C \). The intersection \( A \cap (B \cup C) \) will be the area where circle \( A \) overlaps with the combined area of circles \( B \) and \( C \).

### Step 5: Identify the correct expression
From the properties of sets, we know that:
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]
This means the intersection of \( A \) with the union of \( B \) and \( C \) is equal to the union of the intersection of \( A \) with \( B \) and the intersection of \( A \) with \( C \).

### Conclusion
Thus, the expression \( A \cap (B \cup C) \) can be rewritten as:
\[ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \]