<p>Find the number of functions from set A = {1, 2, 3} to set B = {1, 2, 3, 4} such that the co-domain contains exactly the element 2.</p>
Step-by-Step Solution
Key Concept: Count functions where a specific element (here, 2) must appear in the range by considering all possible co-domains containing that element and using inclusion-exclusion.
<p><strong>Step 1:</strong> The functions must have 2 in their range (co-domain), meaning 2 must be in the image.</p><p><strong>Step 2:</strong> The possible co-domains are those containing 2: {2}, {1,2}, {2,3}, {2,4}.</p><p><strong>Step 3:</strong> For co-domain {2}: all three elements map to 2, giving 1 function.</p><p><strong>Step 4:</strong> For each of the 3 co-domains with 2 and one other element: use surjective count = <span>$2^3 - 2 = 6$</span> functions (total functions minus those missing one element).</p><p><strong>Step 5:</strong> Total = <span>$1 + 3 \times 6 = 19$</span>.</p>
Correct Answer: 19