For any three sets A,B, and `C,(A-B)nn(C-B)` is equal to (i) `A-(BnnC)` (ii) `(A-C)nnB` (iii) `(AnnC)-B` (iv) `(A-B)nnC`

Options

A. `A-(BnnC)`
B. `(A-C)nnB`
C. `(AnnC)-B`
D. `(A-B)nnC`

Answer

To solve the problem, we need to simplify the expression \((A - B) \cap (C - B)\) and determine which of the given options it is equal to. Let's break this down step by step.

### Step 1: Rewrite the Expression
We start with the expression:
\[ (A - B) \cap (C - B) \]

### Step 2: Use the Definition of Set Difference
Recall that the set difference \(A - B\) can be expressed in terms of intersection with the complement:
\[ A - B = A \cap B' \]
where \(B'\) is the complement of set \(B\). Similarly, we can express \(C - B\):
\[ C - B = C \cap B' \]

### Step 3: Substitute into the Expression
Now we can substitute these definitions into our original expression:
\[ (A - B) \cap (C - B) = (A \cap B') \cap (C \cap B') \]

### Step 4: Apply the Associative Property of Intersection
Using the associative property of intersection, we can combine the terms:
\[ (A \cap B') \cap (C \cap B') = A \cap C \cap B' \]

### Step 5: Rewrite the Expression
Now we can rewrite this expression using the definition of set difference:
\[ A \cap C \cap B' = A \cap C - B \]

### Conclusion
Thus, we have simplified \((A - B) \cap (C - B)\) to:
\[ A \cap C - B \]
This matches option (iii) from the choices given.

### Final Answer
The correct option is:
(iii) \((A \cap C) - B\)

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