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.
1. S={(a,b)∣a∈Z+, b∈Z+, and a∣b} S={(a,b)∣a∈Z+, b∈Z+, and a∣b}
2. S={(a,b)∣a∈Z+, b∈Z+, and 3∣a+b} S={(a,b)∣a∈Z+, b∈Z+, and 3∣a+b}
3. S={(a,b)∣a∈Z+, b∈Z+, and a+b is odd} S={(a,b)∣a∈Z+, b∈Z+, and a+b is odd}
a) (1,2)∈S(1,2)∈S, (2,1)∈S(2,1)∈S and if (a,b)∈S(a,b)∈S then (a+2,b)∈S(a+2,b)∈S, (a,b+2)∈S(a,b+2)∈S and (a+1,b+1)∈S(a+1,b+1)∈S
b) (1,2)∈S (1,2)∈S, (2,1)∈S(2,1)∈S and if (a,b)∈S(a,b)∈S then (a+3,b)∈S(a+3,b)∈S, (a,b+3)∈S(a,b+3)∈S, (a+1,b+2)∈S(a+1,b+2)∈S and (a+2,b+1)∈S(a+2,b+1)∈S
c) (1,1)∈S (1,1)∈S and if (a,a)∈S(a,a)∈S then (a+1,a+1)∈S(a+1,a+1)∈S and if (a,b)∈S,(a,b)∈S, then (a,b+a)∈S