(b) How many different functions are there from a set with 8 elements to a set with 5 elements?

To determine the total number of functions from a set $A$ with $n = 8$ elements to a set $B$ with $m = 5$ elements:

• Each element of $A$ can be mapped to any one of the 5 elements of $B$.
• Since the choice for each of the 8 elements of $A$ is independent, the total number of functions is:

$m^n = 5^8$

Calculation:

$5^8 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 390,625$

Total number of functions = $390,625$.

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How many of them are one-to-one?

A function is one-to-one (injective) if no two elements in $A$ are mapped to the same element in $B$. For this to happen:

1. The size of $A$ ($n = 8$) must not exceed the size of $B$ ($m = 5$). In this case, $n > m$, so it is not possible to create a one-to-one function.

$\text{One-to-one functions = 0 (since } n > m\text{).}$

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Final Answers:

1. Total functions: $390,625$.
2. One-to-one functions: $0$.

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Let me know if you'd like a more detailed explanation!

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Related Questions to Explore:

1. How does the size of the codomain affect the total number of functions?
2. What happens to the number of functions if the domain and codomain sizes are equal?
3. How many surjective functions can be defined in this case?
4. What if the codomain had 6 elements instead of 5?
5. Can you generalize this calculation to arbitrary $n$ and $m$?

Tip: When working with injective functions, always check if $n \leq m$ first—it's a necessary condition!