Total Number of Functions from $A$ to $B$:

A function from set $A$ to set $B$ is defined by assigning each element in $A$ to an element in $B$. Since $A$ has 3 elements and $B$ has 3 elements, for each element in $A$, we have 3 possible choices from $B$.

Thus, the total number of functions is calculated as: $|B|^{|A|} = 3^3 = 27.$

Number of One-to-One (Injective) Functions:

For a function to be one-to-one (injective), each element in $A$ must map to a distinct element in $B$. Therefore, we are selecting 3 distinct elements from $B$ to assign to the 3 elements in $A$. This is simply the number of permutations of 3 elements from $B$, which is given by: $P(3, 3) = 3! = 6.$

Number of Onto (Surjective) Functions:

A function is onto (surjective) if every element of $B$ has at least one preimage in $A$. To count the number of onto functions from $A$ to $B$, we use the inclusion-exclusion principle. The number of functions where all elements of $B$ are used at least once is given by: $|S_3| = 3! = 6.$ This happens because $A$ and ( B \ can ans