Give a recursive definition of each of these sets of ordered pairs of positive integers. [Hint: Plot the points in the set in the plane and look for lines containing points in the set.
$S = \left\{{(a,b) \mid a \in Z^+, b \in Z^+ \text{ and } a + b \text{ is odd}}\right\}$
Does this work?
$(1,2) \in A$
$(2,1) \in A$
if $(a, b) \in A$ s.t. $a = 2k + 1$ where $k$ is a constant and a and b are integers, then $(a + 2, b) \in A$
if $(a, b) \in A$ s.t. $a = 2k + 1$ then $(a, b + 2) \in A$
I think the above gives me all the pairs where $a$ is odd $b$ is even.
if $(a, b) \in A$ s.t. $a = 2m$ (where $m$ is a constant) then $(a + 2, b) \in A$
if $(a, b) \in A$ s.t. $a = 2m$ then $(a, b + 2) \in A$
I think these two conditionals give me all the pairs where $a$ is even and $b$ is odd.