CAT 2025 Slot 3 QA Question 16: Consider two sets $A = \{2, 3, 5, 7, 11, 13\}$ and $B = \{1, 8, 27\}$. Let $f$ be a function from $A$ to $B$ such that for every element $b$ in $B$ there is at least one element $a$ in $A$ such that $

Question

Consider two sets $A = \{2, 3, 5, 7, 11, 13\}$ and $B = \{1, 8, 27\}$. Let $f$ be a function from $A$ to $B$ such that for every element $b$ in $B$ there is at least one element $a$ in $A$ such that $

Accepted Answer

The condition says that every element of $B$ must be used at least once. So, $f$ must be onto.
Total functions from a $6$-element set to a $3$-element set are $3^6$.
Use inclusion-exclusion to remove functions that miss at least one element of $B$.
Number of onto functions = $3^6 - \binom{3}{1}	ime