Question 47
Let $k \in \mathbb{N}$ and define the sets
$$A_k = \bigcup_{i=0}^{k+1} \{(m, n) \in \mathbb{N} \times \mathbb{N} \mid m \ge 2i,\ n \ge 2(k-i) + 1\},$$
$$B_k = \bigcup_{i=-1}^{k} \{(m, n) \in \mathbb{N} \times \mathbb{N} \mid m \ge 2i + 1,\ n \ge 2(k-i)\}.$$
Prove that
$$A_k \cup B_k = \{(m, n) \in \mathbb{N} \times \mathbb{N} \mid m + n \ge 2k + 1\},$$
$$A_k \cap B_k = \{(m, n) \in \mathbb{N} \times \mathbb{N} \mid m + n \ge 2k + 2\}.$$
$$A_k = \bigcup_{i=0}^{k+1} \{(m, n) \in \mathbb{N} \times \mathbb{N} \mid m \ge 2i,\ n \ge 2(k-i) + 1\},$$
$$B_k = \bigcup_{i=-1}^{k} \{(m, n) \in \mathbb{N} \times \mathbb{N} \mid m \ge 2i + 1,\ n \ge 2(k-i)\}.$$
Prove that
$$A_k \cup B_k = \{(m, n) \in \mathbb{N} \times \mathbb{N} \mid m + n \ge 2k + 1\},$$
$$A_k \cap B_k = \{(m, n) \in \mathbb{N} \times \mathbb{N} \mid m + n \ge 2k + 2\}.$$