To calculate the total number of many-one functions from set $A$ to set $B$, let's break it down step-by-step.

Definitions:

1. A many-one function is a function where multiple elements in the domain ($A$) can map to the same element in the codomain ($B$).

2. The total number of functions $f: A \to B$ is given by: $|B|^{|A|}$ where $|B|$ is the size of $B$, and $|A|$ is the size of $A$.

3. Subtracting the one-to-one (injective) functions from the total number of functions gives the total many-one functions, because one-to-one functions are a subset of functions.

Steps:

1. Compute the total number of functions:

$|B| = 5, \, |A| = 4$ $\text{Total functions} = 5^4 = 625$

2. Compute the total number of one-to-one functions:

A one-to-one function requires all elements of $A$ to map to distinct elements of $B$. The number of injective functions is: $P(|B|, |A|) = \frac{|B|!}{(|B| - |A|)!} = \frac{5!}{(5-4)!} = \frac{120}{1} = 120$

3. Subtract one-to-one functions from total functions:

$\text{Many-one functions} = \text{Total functions} - \text{One-to-one functions}$ $\text{Many-one functions} = 625 - 120 = 505$

Final Answer:

The total number of many-one functions from $A$ to $B$ is 505.

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Do you want a deeper explanation or further assistance? Here are some related questions:

1. How do we generalize the formula for $m$-to-$n$ many-one functions?
2. Can you explain the difference between one-to-one and many-one functions with examples?
3. How would the result change if $|B| < |A|$?
4. How do these concepts apply to real-world problems in mathematics?
5. Can you illustrate the mapping of elements for one-to-one and many-one functions?

Tip: Always remember that for many-one functions, overlapping mappings (multiple domain elements sharing a codomain value) are allowed!