<p>Total symmetric relations on <span class="math-inline">\(A\)</span> (<span class="math-inline">\(|A|=n\)</span>):</p>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Diagonal entries: <span class="math-inline">$n$</span> free choices. Upper triangle: <span class="math-inline">$n(n-1)/2$</span> free pairs (each determines its mirror). Total: <span class="math-inline">$2^{n+n(n-1)/2}=2^{n(n+1)/2}$</span>.</p><div class="key-concept"><strong>Key Concept:</strong> Symmetry identifies (i,j) with (j,i) — count only one triangle plus diagonal</div></div>
Correct Answer: 2^(n(n+1)/2)