True or false: Please label each of the following statements as either true or false and

Chapter 54, Problem 54.9

(choose chapter or problem)

True or false: Please label each of the following statements as either true or false and then give a proof. a. Let x and y be elements of a poset. It must be the case that exactly one of the following is true: x < y, x D y, or x > y. b. Let x and y be elements of a poset and suppose there is a chain that contains both x and y. Then it must be the case that exactly one of the following is true: x < y, x D y, or x > y. c. Let C and D be chains in a poset. Then C [ D is also a chain. d. Let C and D be chains in a poset. Then C \ D is also a chain. e. Let A and B be antichains in a poset. Then A [ B is also an antichain. f. Let A and B be antichains in a poset. Then A \ B is also an antichain. g. Let A be an antichain and C be a chain in a poset. Then A \ C must be empty. h. Two points in a Hasse diagram (representing two elements of a poset) can never be joined by a horizontal line segment. i. Let A be a set of elements in a poset. If no two elements of A are joined by a curve in the Hasse diagram, then A is an antichain. j. Let A be a set of elements in a poset. If A is an antichain, then no two elements of A are joined by a curve in the Hasse diagram.