Question

The shaded region in the following Venn Diagram represents:
\begin{center} \includegraphics[width=0.7\textwidth]{02.jpeg} \end{center}

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

Hint:

When a shaded zone is "this overlap but not inside the third set", write it as an intersection with the complement: \((A\cap B)\cap C^{\complement}\). Then test options using identities like \(X\cap(Y\cup Z)=(X\cap Y)\cup(X\cap Z)\).

Answer 1

The Correct Option is D

Step 1: Translate the shaded part into set language.
From the diagram, the shading is exactly the overlap of \(A\) and \(B\) \emph{excluding} the portion that also lies in \(C\).
Hence the region is \((A \cap B)\) minus \(C\), i.e. \((A \cap B) \cap C^{\complement}\). Step 2: Compare with each option.
(a) \(A \cup (B \cup C) = A \cup B \cup C\): the entire union of all three sets — far larger than the shaded lens.
(b) \(A \cup (B \cap C)\): includes all of \(A\) plus the overlap \(B \cap C\) — again much larger.
(c) \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\): includes the \(A\cap B\) lens \emph{and} the \(A\cap C\) cap, so it contains points the diagram does not shade.
None of (a), (b), (c) equals \((A \cap B) \cap C^{\complement}\). Step 3: Conclude.
Therefore the correct description is \((A \cap B) \cap C^{\complement}\), which is not listed.
\[ \boxed{\text{None of these}} \]