Suppose that R is an order relation on a set S and that T is a subset of S. Prove the following: (a) If T has a minimum element, it is unique. (b) If q is a minimum element, then q is minimal. (c)...

Suppose that R is an order relation on a set S and that T is a subset of S. Prove the following:

(a) If T has a minimum element, it is unique.

(b) If q is a minimum element, then q is minimal.

(c) Prove that b covers a , b is a minimal element of {x = S \ {a}: aRx}.

Suppose that R is an order relation on a set S and that T is a subset of S. Prove the following:

(a) If T has a maximum element, it is unique.

(b) If q is a maximum element, then q is maximal.

(c) If T is finite, T must have at least one maximal element.

(d) What is the minimum upper bound for the subset in?

Chapter 8