Final answer:
The set S contains all ordered pairs of integers where both integers are either both even or both odd, starting from (1, 2) and (2, 1). Recursive operations maintain the parity of each number, resulting in a pattern akin to a checkerboard when visualized.
Step-by-step explanation:
The question presented asks for a simple description of the set S of ordered pairs of integers. By analyzing the recursive definition provided, we can deduce that set S consists of all pairs (m, n) where m and n start at 1 and 2, respectively, and can be increased by 2 or by 1 synchronously. Whenever we increase one coordinate by 2, the other can stay the same, or it can also increase by 2, thus making both even or odd. However, when we increase each coordinate by 1, we get that both will have the same parity (odd or even) as they started with.
The simple description of the set S, considering the recursive definition, is that it contains all ordered pairs of integers where both integers are either both even or both odd, starting from (1, 2) and (2, 1). This is because the operations given in the definition maintain the parity of each number in the pair. Consequently, S includes all pairs (m, n) where m and n have the same parity (both even or both odd), with m starting at 1 and n starting at 2.
This means if you take any ordered pair (m, n) from the set, m and n will either both be odd or both be even. For instance, starting from (1, 2), applying the rule '(m+1, n+1)' gives us (2, 3), which becomes a new starting point for generating more pairs. Thus, set S could be visually represented on a coordinate grid where the dots form a checkerboard pattern, with dots for pairs with both members being odd in one color and those with both members being even in another color.