To solve both parts of this problem, we need to utilize combinatorial and algebraic techniques. Here’s a breakdown of each part:

Part (i)

Prove that: $\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0$

This sum represents the alternating sum of binomial coefficients. We can prove this using the Binomial Theorem.

Approach:

1. Using the Binomial Theorem: The binomial theorem states that for any integer $n$, $(1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k.$
2. To find the sum in the question, substitute $x = -1$: $(1 + (-1))^n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k.$
3. On the left side, $1 + (-1) = 0$, so this becomes: $0^n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k.$
4. For any positive integer $n$, $0^n = 0$, which gives: $\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0.$

Thus, this proves the required result for part (i).

Part (ii)

Prove that the set $\{1, \dots, n\}$ has exactly as many subsets with an odd number of elements as subsets with an even number of elements.

Approach:

This result can be proved by using a combinatorial argument or by algebraic reasoning. Here is a combinatorial reasoning:

1. Consider all subsets of a set with $n$ elements. There are $2^n$ total subsets.
2. Each subset can either have an odd number of elements or an even number of elements.
3. From part (i), we know that: $\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0.$
4. Notice that this sum can be rewritten as: $\sum_{\text{even } k} \binom{n}{k} - \sum_{\text{odd } k} \binom{n}{k} = 0.$
5. This implies that: $\sum_{\text{even } k} \binom{n}{k} = \sum_{\text{odd } k} \binom{n}{k}.$
6. Therefore, the number of subsets with an even number of elements is equal to the number of subsets with an odd number of elements.

This completes the proof for part (ii).

Would you like further details on either proof, or have questions on specific steps?

Here are some related questions to further explore:

1. How does the Binomial Theorem apply to different values of $x$ in combinatorial proofs?
2. What other proofs exist for the alternating sum of binomial coefficients?
3. How can generating functions be used to prove combinatorial identities?
4. What are some applications of even and odd subset properties in probability?
5. How would the results change if subsets of specific sizes were excluded?

Tip: In combinatorics, substituting specific values into known theorems (like setting $x = -1$ in the Binomial Theorem) often simplifies or reveals new insights into sums and identities.