Sets, Relations & Functions
Venn diagrams
Grade 11
Question:
<p>The shaded region in the given figure is</p><p><em>(The figure shows three overlapping circles A, B, C where the region inside A but outside B and C is shaded.)</em></p>
<p>\(A \cap (B \cup C)\)</p>
<p>\(A \cup (B \cap C)\)</p>
<p>\(A \cap (B - C)\)</p>
<p>\(A - (B \cup C)\)</p>
Step-by-Step Solution
Key Concept: The shaded region represents elements that belong to set A but are excluded from both B and C simultaneously. This is expressed as A − (B ∪ C) or equivalently A ∩ B' ∩ C'.
<p><strong>Step 1:</strong> Identify what region is shaded: The part inside circle A that lies outside both circles B and C.</p><p><strong>Step 2:</strong> Express the condition: Elements must satisfy: (in A) AND (not in B) AND (not in C).</p><p><strong>Step 3:</strong> Write in set notation: This is A ∩ (B ∪ C)' or equivalently A − (B ∪ C) or A ∩ B' ∩ C'.</p><p><strong>Step 4:</strong> Verify: Using De Morgan's law, (B ∪ C)' = B' ∩ C', confirming A ∩ B' ∩ C' = A − (B ∪ C).</p><p>∴ Answer: <strong>D</strong> (which represents A − (B ∪ C) or A ∩ (B ∪ C)')</p>
Correct Answer: D