Let A, B and C be three sets, then which of the following is true?

Options

A. `A xx (B-C) = (A-B) xx (A-C)`
B. `A xx (B - C) = A xx B - A xx C`
C. `A xx (B cap C) = B cap C xx A`
D. `A xx (B cap C) = (B cap C)xx A`

Answer

To solve the problem regarding the sets A, B, and C, we need to analyze the options provided and determine which statement is true. Let's break it down step by step.

### Step 1: Understanding Cartesian Product
The Cartesian product of two sets A and B, denoted as A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

### Step 2: Analyzing the Options
We need to evaluate the options given in the question. The options are likely related to properties of Cartesian products and set operations.

### Step 3: Associative Law of Cartesian Products
The associative law states that for any sets A, B, and C:
- (A × B) × C = A × (B × C)

This means that the way we group the sets when taking the Cartesian product does not affect the result.

### Step 4: Evaluating the Expression A × (B - C)
Let's evaluate the expression A × (B - C):
- B - C is the set of elements that are in B but not in C.
- Therefore, A × (B - C) would consist of all pairs (a, b) where a ∈ A and b ∈ (B - C).

### Step 5: Evaluating the Expression (A × B) - (A × C)
Now, consider the expression (A × B) - (A × C):
- A × B consists of all pairs (a, b) where a ∈ A and b ∈ B.
- A × C consists of all pairs (a, c) where a ∈ A and c ∈ C.
- The difference (A × B) - (A × C) will give us all pairs (a, b) where a ∈ A and b ∈ B, but we exclude those pairs where b is also in C.

### Step 6: Conclusion
From the analysis, we can see that:
- A × (B - C) = (A × B) - (A × C)

Thus, the statement that is true is that the Cartesian product distributes over set difference.

### Final Answer
The correct statement is:
A × (B - C) = (A × B) - (A × C)