Sets, Relations & Functions
Cartesian Product
Grade 11
Question:
<p>If <span class="latex">n(A) = 4</span>, <span class="latex">n(B) = 3</span>, <span class="latex">n(A \times B \times C) = 24</span>, then <span class="latex">n(C)</span> equals</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) 17</p>
<p>(d) 288</p>
Step-by-Step Solution
Key Concept: The cardinality of a Cartesian product of sets is the product of their individual cardinalities.
<p><strong>Step 1:</strong> Use the property <span class="latex">n(A \times B \times C) = n(A) \cdot n(B) \cdot n(C)</span></p><p><strong>Step 2:</strong> Substitute given values: <span class="latex">24 = 4 \times 3 \times n(C)</span></p><p><strong>Step 3:</strong> Solve for <span class="latex">n(C)</span>: <span class="latex">n(C) = \frac{24}{12} = 2</span></p><p>∴ Answer is (b).</p>
Correct Answer: b