Relations & Functions
Increasing and Decreasing Functions
Grade 12
Question:
<p>Let \(A = \{1, 2, 3, 4, 5\}\) and \(B = \{-2, -1, 0, 1, 2, 3, 4, 5\}\). The number of increasing functions from A to B is</p>
<p>(a) 120</p>
<p>(b) 72</p>
<p>(c) 60</p>
<p>(d) 56</p>
Step-by-Step Solution
Key Concept: An increasing function is equivalent to selecting 5 distinct elements from B in strictly increasing order.
<p><strong>Solution:</strong> For an increasing function from A to B, we need $f(1) < f(2) < f(3) < f(4) < f(5)$. This is equivalent to choosing 5 distinct elements from B in increasing order. The number of such functions is $\binom{8}{5} = 56$.</p>
Correct Answer: D