Final Exam
for Discrete Structures
Bokelavadze, Imerlishvili,
Nadareishvili & Neuhauser
1
Instructions: You have 25 minutes (including upload) to complete this part of the exam.
1. Write down a seven element subset of the set of integers Z.
(1 p)
2. Let f : N → N be a function given by f (n) = 3n. Determine f −1 ({4, 5, 6, 7, 8, 9, 10}).
(1 p)
3. Is the relation
R = {(a, b) ∈ N0 × N0 : a divides b}
on the set of non-negative integers transitive? Explain your answer.
(2 p)
4. How many equivalence relations are there on the set X = {1, −7, −2}? Explain.
(2 p)
5. Write down three integers congruent to 16 modulo 3.
(1 p)
6. Does every subset in (N, ≤) possess an supremum? Explain.
(1 p)
1
Final Exam
for Discrete Structures
Bokelavadze, Imerlishvili,
Nadareishvili & Neuhauser
2
Instructions: You have 25 minutes (including upload) to complete this part of the exam.
7. Compute the following in B(5):
(1 p)
(1, 1, 1, 0, 1)(1, 0, 1, 0, 1) + (0, 1, 0, 1, 0).
8. Construct the Disjunctive Normal Form for the function f : B(3) → {0, 1} given by
(2 p)
f (x, y, z) = (x → ¬z) → (¬y ∧ z).
9. Find the supremum of the set {(1, 0, 1, 1), (0, 0, 1, 1), (1, 0, 0, 1)} in B(4).
(1 p)
10. Does there exist a bijection between the sets {9, 67, 1, 14, π, 1/2} and {π, 9, 67, 15, 1, 1/2}?
Explain your answer.
(1 p)
11. Evaluate the following
(1 p)
4
4 Y
X
j+2
.
2
j=1 i=2 i − 1
12. Rank the following functions from the slowest growing to the fastest growing (that is, using
the relation ≺):
(2 p)
0.01n2 , 2 log n, (11/10)n , 8, 1.3n!, n5 , 2nn , n.
1
Final Exam
for Discrete Structures
Bokelavadze, Imerlishvili,
Nadareishvili & Neuhauser
3
Instructions: You have 25 minutes (including upload) to complete this part of the exam.
13. Sketch the graph G = (V, E) with V = {1, 2, 3, 4, 5} and E = {{2, 3}, {1, 2}, {3, 4}, {4, 5}}.
(1 p)
14. Write down the sequence of degrees for the graph on Figure 1.
(1 p)
Figure 1
15. Sketch a directed graph with 1 sink and 4 sources.
(1 p)
16. Find a minimal spanning tree for the weighted graph on Figure 2. Explain why the tree you
gave is indeed minimal.
(2 p)
5
6
3
5
10
2
10
4
9
9
0
1
11
Figure 2
1
6
5
Final Exam
Discrete Structures
17. The cost matrix is given by the table of distances (in meters) on Figure 3 between different
buildings on KIU campus. Use the “cheapest insertion” algorithm to find a short trip through
every building starting at building E.
(2 p)
K
E
F
G
H
K
E
F
G
H
– 1620 1510 1380 1320
1620
– 110 240 310
1510 110
– 120 210
1380 240 120
–
90
1320 310 210
90
–
Figure 3
18. Is the graph on Figure 1 Hamiltonian? Explain your answer.
2
(1 p)
Final Exam
for Discrete Structures
Bokelavadze, Imerlishvili,
Nadareishvili & Neuhauser
4
Instructions: You have 30 minutes (including upload) to complete this part of the exam.
19. Show that division operation ÷ : (R \ {0}) × (R \ {0}) → (R \ {0}} is not associative. (1 p)
20. Show that 4 ∈ h 91 i, where h 19 i is the subgroup of (Q, +) generated by 91 .
(1 p)
21. Find all zero divisors in Z14 .
(1 p)
22. Solve the following equation in S5 :
(2 p)
1 2 3 4 5
1 4 2 3 5
!
◦x=
1 2 3 4 5
1 5 2 4 3
!
.
23. Decompose
(1 p)
1 2 3 4 5
5 3 2 1 4
!
into a product of disjoint cycles.
24. Give an example of a non-trivial ideal I in Z10 (that is, I 6= Z10 and I 6= {0}).
(1 p)
25. Write down a Cayley table for multiplication for a four-element field.
(1 p)
Bonus
26. Prove that for any finite group G, the number of elements (order) in every subgroup of G
divides the number of elements (order) of G.
(4 p)
1