Hi, I have a discrete and combinatorial math question.

I need your help.

Thank you so much.

4. (4 marks) Is it true that in any finite poset? every longest Chain contains an element of every largest antichain'? Justify your answer. .5. Let X be a set With lXI : 71.. (a) (1 mark) How many longest Chains are there in (73(X), g)? [No justication required.) (b) (2 marks) Find a partition of (73(X) g) into |X| + 1 antiehains. [No justication re quired.) (C) (2 marks) Given any set A E X (t)? how many longest Chains of (73(X), Q) contain A? (d) (3 marks) Let A be an antichain in ('P(X), g)? and let at denote the number of elements of A of size t. Prove that n 71 Z at t!(-n t)! S n! or, equivalently, Z (:t g 1. t:0 k:0 (If) 6. (Bonus 3 marks) Prove that (73([n.]), g) can be partitioned into (lgl) Chains. 2 2.1 Posets A relation _ on P is called a partial order if it satisfies the three conditions: . (