Attempted answers:

MD 1. \\(\Delta\\) is monotonic, because \\(a \leq b \to (a,a) \leq (b,b)\\), by definition of \\(\leq_{A\times A}\\)

MD 2. By the method in lecture 6, r(y,z) = least upper bound of X = \\(\\{x : \Delta (x) \leq (y,z) \\}\\). Since X is the set of elements of A less than min(y,z), r(y,z) is min(y,z).

MD 3. By duality, l(y,z) = max(y,z)

MD 4.

r: least upper bound of X = \\(\\{x : \Delta (x) \leq (y,z) \\}\\): least common multiple

l: greatest lower bound of X = \\(\\{x : \Delta (x) \geq (y,z) \\}\\) : greatest common divisor

I think I've been sloppy and got some of this flipped - to be fixed later.

MD 1. \\(\Delta\\) is monotonic, because \\(a \leq b \to (a,a) \leq (b,b)\\), by definition of \\(\leq_{A\times A}\\)

MD 2. By the method in lecture 6, r(y,z) = least upper bound of X = \\(\\{x : \Delta (x) \leq (y,z) \\}\\). Since X is the set of elements of A less than min(y,z), r(y,z) is min(y,z).

MD 3. By duality, l(y,z) = max(y,z)

MD 4.

r: least upper bound of X = \\(\\{x : \Delta (x) \leq (y,z) \\}\\): least common multiple

l: greatest lower bound of X = \\(\\{x : \Delta (x) \geq (y,z) \\}\\) : greatest common divisor

I think I've been sloppy and got some of this flipped - to be fixed later.