To prove that $A = C$, we need to show that every element of set $A$ is in set $C$, and vice versa.

Definitions of Sets $A$ and $C$

1. Set $A$: $A = \{n \in \mathbb{Z} \mid n = 4 - k, \text{ for some } k \in \mathbb{N} \text{ and } 5 \mid k \}$ This means $n$ is of the form $4 - k$ where $k$ is a natural number and a multiple of 5, i.e., $k = 5m$ for some $m \in \mathbb{N}$.

2. Set $C$: $C = \{m \in \mathbb{Z} \mid m = 24 - 5k, \text{ for some } k \in \mathbb{N} \text{ and } k \geq 4 \}$ This means $m$ is of the form $24 - 5k$ where $k$ is a natural number and $k \geq 4$.

Proof that $A \subseteq C$

1. Let $n \in A$. By definition, $n = 4 - k$ where $k = 5m$ for some $m \in \mathbb{N}$.

2. Substituting $k = 5m$: $n = 4 - 5m$

3. To express $n$ in the form required for set $C$: $n = 24 - 5(k')$ where $k'$ needs to be determined.

4. Set $4 - 5m = 24 - 5k'$. Solving for $k'$: $24 - 5k' = 4 - 5m$ $20 = 5k' - 5m$ $4 = k' - m$ $k' = m + 4$

5. Since $m \in \mathbb{N}$, $k' = m + 4 \geq 4$ because $m \geq 1$. Therefore, $k' \in \mathbb{N}$ and $k' \geq 4$.

6. Thus, $n = 4 - 5m = 24 - 5k'$ with $k' \geq 4$, implying $n \in C$. Therefore, $A \subseteq C$.

Proof that $C \subseteq A$

1. Let $m \in C$. By definition, $m = 24 - 5k$ where $k \geq 4$ and $k \in \mathbb{N}$.

2. Set $k = 4 + m'$ where $m' \in \mathbb{N}$ to satisfy $k \geq 4$.

3. Substituting $k = 4 + m'$: $m = 24 - 5(4 + m')$ $m = 24 - 20 - 5m'$ $m = 4 - 5m'$

4. Since $m' \in \mathbb{N}$, let $m' = m$ (to maintain the natural number property). Thus, $m = 4 - 5m' = 4 - 5m$ with $5m \in \mathbb{N}$ as required by $A$.

5. Therefore, $m \in A$. This implies $C \subseteq A$.

Conclusion

Since we have shown that $A \subseteq C$ and $C \subseteq A$, it follows that:

$A = C$

Do you have any questions or would you like more details?

Related Questions:

1. How would this proof change if we altered the conditions on $k$?
2. Can we use a different method, such as set notation manipulation, to prove $A = C$?
3. How can we visualize sets $A$ and $C$ on a number line?
4. How does changing the multiple constraint from $5$ to another number affect the sets?
5. What happens to the sets if we allow $k$ to be in $\mathbb{Z}$ instead of $\mathbb{N}$?

Tip: Always clearly define the elements of each set before attempting to prove equality; this ensures you accurately capture the conditions needed for membership in each set.