Which is the simplified representation of `(A' cap B' cap C) cup (B cap C ) cup (A cap C)` where A,B and C are subsets of set X ?

Options

A. A
B. B
C. C
D. `X cap (A cup B cup C)`

Explanation

To simplify the expression \((A' \cap B' \cap C) \cup (B \cap C) \cup (A \cap C)\), where \(A\), \(B\), and \(C\) are subsets of a universal set \(X\), we can follow these steps:

### Step 1: Rewrite the expression
The original expression is:
\[ (A' \cap B' \cap C) \cup (B \cap C) \cup (A \cap C) \]

### Step 2: Apply De Morgan's Law
Using De Morgan's Law, we can rewrite \(A' \cap B'\) as \((A \cup B)'\). Thus, we have:
\[ ((A \cup B)' \cap C) \cup (B \cap C) \cup (A \cap C) \]

### Step 3: Factor out \(C\)
Notice that \(C\) is a common factor in all parts of the union. We can factor \(C\) out:
\[ C \cap ((A \cup B)' \cup B \cup A) \]

### Step 4: Simplify the expression inside the parentheses
Now simplify the expression inside the parentheses:
\[ (A \cup B)' \cup B \cup A \]
This can be simplified further. The expression \((A \cup B)'\) means everything not in \(A\) or \(B\). Therefore, when we take the union with \(B\) and \(A\), we can conclude that:
\[ (A \cup B)' \cup B \cup A = X \]
where \(X\) is the universal set.

### Step 5: Final simplification
Now substituting back, we have:
\[ C \cap X \]
Since \(C\) is a subset of \(X\), this simplifies to:
\[ C \]

Thus, the simplified representation of the original expression is:
\[ \boxed{C} \]