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Question
Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.
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Solution
Formula:
When two sets A and B have m and n elements respectively, then the number of onto functions from A to B is
\[\binom{ \sum\nolimits_{r = 1}^n \left( - 1 \right)^r n C_r r^m , \text{if m} \geq n}{\text{o if m} < n }\]
Here, number of elements in A = 4 = m
Number of elements in B = 2 = n
So, m > n
Number of onto functions
\[= \sum\nolimits_{r = 1}^2 \left( - 1 \right)^r 2 C_r r^4 \]
\[ = \left( - 1 \right)^1 2 C_1 1^4 + \left( - 1 \right)^2 2 C_2 2^4 \]
\[ = - 2 + 16\]
= 14
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