Allow us to say that a limited incomplete request (A, v) is tree-like if, for each a ∈ A, the arrangement (of
its ancestors) {x ∈ A | x v a ∧ x 6= a} either is unfilled or has a novel maximal
component. Equally, pictorially, this holds when the Hasse chart of A comprises
of at least one trees.
State which of the accompanying relations on the whole numbers {1, 2, . . . , 10} are tree-like
halfway orders and give a one-sentence avocation.
(a) R where xRy ⇔ x = y
(b) R where xRy ⇔ x 6 y (here 6 is the typical requesting on numbers)
(c) R where xRy ⇔ x partitions precisely into y
(d) R where xRy ⇔ x = y or x is the best prime variable of y
[8 marks]
To count the number C(n) of tree-like incomplete orders of n components, accept
A = {1, 2, . . . , n} and afterward place each and every component I into a Hasse outline
beginning from 1 and with the end goal that no later component j > I is put to such an extent that j v I.
Show that, gave n > 1, we have C(n) = f(n, C(n − 1)) and give the capacity
f(n, m). Give a base case and accordingly tackle the repeat for C(n). [12 marks]
5 [TURN OVER
CST.97.1.6
8 Discrete Mathematics
Let A be a set and R a connection on A; likewise compose Rk
for the standard k-crease piece
of R, for example R1 = R, Rk+1 = R ◦ Rk
. Let t(R) be the littlest connection which is
transitive and has R ⊆ t(R), comparatively let u(R) = S∞
k=1 Rk

1. [CLO4, A3, PLO5] A local hospital operates FIVE (5) clinics and employed SEVEN (7) chief doctors. The rule in the local hospital is only ONE (1) chief doctor can be in- charge on any clinic on a single day and each doctor cannot work more than TWO (2) days in a row. Due to current COVID-19 pandemic, the local hospital utilized all clinics every day. Here, Genetic Algorithm (GA) is used to result in the best combination of chief doctors in-charged on any particular day. Answer the following questions based on given scenario. (a) State the number of chromosome(s) that represent an individual in Genetic Algorithm. [2 mark] (b) Suggest an objective function for the stated problem in Question 1. [2 mark] (c) With your knowledge in Genetic Algorithm, how many solutions provided in this scenario using the algorithm. [3 marks] (d) Provide your discussion whether the use of Genetic Algorithm to solve the problem in the local hospital is necessary. Include also in your discussion, what if the local hospital operated with more chief doctors and clinics. [8 marks]