Sets, Relations & Functions
Set operations
Grade 11
Question:
<p>For sets <em>A</em> and <em>B</em>, \((A \cup B)' \cup (A' \cap B)\) equals</p>
<p>\(A'\)</p>
<p>\(B'\)</p>
<p>\(A\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Use De Morgan's laws systematically: (A ∪ B)' = A' ∩ B', then recognize that (A' ∩ B') ∪ (A' ∩ B) factors to A' ∩ (B' ∪ B) = A'.
<p><strong>Step 1:</strong> Apply De Morgan's law to (A ∪ B)'<br>$(A ∪ B)' = A' ∩ B'$</p><p><strong>Step 2:</strong> Substitute into the original expression<br>$(A' ∩ B') ∪ (A' ∩ B)$</p><p><strong>Step 3:</strong> Factor out A' using distributive property<br>$= A' ∩ (B' ∪ B)$</p><p><strong>Step 4:</strong> Simplify B' ∪ B<br>$B' ∪ B = U$ (Universal set, since every element is either in B or not in B)</p><p><strong>Step 5:</strong> Final simplification<br>$A' ∩ U = A'$</p><p>∴ Answer: <strong>A'</strong> (complement of A)</p>
Correct Answer: A