Sets, Relations & Functions
Set operations
Grade 11
Question:
<p>Which is the simplified representation of \((A' \cap B' \cap C) \cup (B \cap C) \cup (A \cap C)\) where <em>A</em>, <em>B</em> and <em>C</em> are subsets of set <em>X</em>?</p>
<p>\(A\)</p>
<p>\(B\)</p>
<p>\(C\)</p>
<p>\(X \cap (A \cup B \cup C)\)</p>
Step-by-Step Solution
Key Concept: Use De Morgan's laws and distributive properties to factor out common elements. Notice that A' ∩ B' ∩ C means elements in C but not in A or B, which can be combined with other C-containing terms by factoring C out.
<p><strong>Step 1:</strong> Observe that all three terms in the union involve set C: (A' ∩ B' ∩ C) ∪ (B ∩ C) ∪ (A ∩ C)</p><p><strong>Step 2:</strong> Factor out C using the distributive property: C ∩ [(A' ∩ B') ∪ B ∪ A]</p><p><strong>Step 3:</strong> Simplify the inner bracket. Note that (A' ∩ B') ∪ B ∪ A contains:
<br/>• Elements from B (which covers B ∩ C)
<br/>• Elements from A (which covers A ∩ C)
<br/>• Elements in neither A nor B (which covers A' ∩ B' ∩ C)
<br/>Therefore: (A' ∩ B') ∪ B ∪ A = X (the universal set, since every element is either in A, in B, or in neither)</p><p><strong>Step 4:</strong> Thus: C ∩ X = C</p><p>∴ <strong>Answer: C</strong></p>
Correct Answer: C