King Saud University
MATH 151 (Discrete Mathematics)
Department of Mathematics
Final Exam
1st Semester 1437-1438 H
Duration: 3 Hours
Name:
Sequence Number:
Teacher's Name:
Section:
Note: The exam consists of 11 pages.
Question
Mark
Question I
Question II
Question III
Question IV
Question V
Total
1
Question
1
2
3
4
5
6
7
8
9
10
11
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Answer
Question I:
Choose the correct answer, then fill in the table above:
(1) ∃! 𝑥 ∈ ℤ 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡
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𝑥
𝜖 ℤ is
(a) True
(b) False
(c) None of the previous
____________________________________________________________________________________
𝑝
𝑝→𝑞
(2) The argument ¬𝑞 ∨ 𝑟 is
___________
∴𝑟
(a) Valid
(b) Invalid
(c) None of the previous
____________________________________________________________________________________
(3) If 𝐴 𝑎𝑛𝑑 𝐵 are subsets of the universal set 𝑈, then 𝐴 − (𝐴 − 𝐵) =
(a) 𝐴
(b) 𝐴 ∪ 𝐵
(c) 𝐴 ∩ 𝐵
(d) None of the previous
_____________________________________________________________________________________
(4) |{𝑎, {𝑎}, {𝑎, {𝑎}}}| =
(a) 8
(b) 3
(c) 4
(d) None of the previous
_____________________________________________________________________________________
(5) If 𝑅 = {(𝑎, 𝑏)|𝑎 > 𝑏}, is a relation on ℤ, then the symmetric closure of R is
(a) {(𝑎, 𝑏)|𝑎 < 𝑏}
(b) {(𝑎, 𝑏)|𝑎 = 𝑏}
(c) {(𝑎, 𝑏)|𝑎 ≠ 𝑏}
(d) None of the previous
_________________________________________________________________________________
(6) Let 𝑈 = {1,2,3} be a nonempty set and let 𝑅 = {(𝐴, 𝐵)|𝐴 ⊆ 𝐵, 𝐴, 𝐵 ∈ 𝑃(𝑈)} be the inclusion
partially ordered relation. Then {2,3} 𝑎𝑛𝑑 {2} are
(a) Comparable
(b) Incomparable
(c) None of the previous
2
Total
represents the poset on {−1,0,1,2} given by:
(7) The Hasse graph
(b) {(𝑥, 𝑦)|𝑥 ≤ 𝑦})
(a){(𝑥, 𝑦)|𝑥 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 𝑦}
(c) {(𝑥, 𝑦)|𝑥 ≥ 𝑦}
(d) None of the previous
_________________________________________________________________________________
(8) For the equivalence relation on ℤ defined by 𝑎𝑅𝑏 ⟺ 𝑎 ≡ 𝑏(𝑚𝑜𝑑5), [6] =
(a) {… , −4,1,6,11,16, … }
(b) {… , −12, −6,0,6,12, … }
(c) {… , −8, −3,2,7,12, … }
(d) None of the previous
________________________________________________________________________________
(9) There exists a graph with vertices of degrees
(a) 1,2,3,3
(b) 3,2,4,6
(c) 1,2,2,3
(d) None of the previous
___________________________________________________________________________________
(10) The graph 𝐶4 is
(a) Not connected
(b) Not Bipartite
(c) Bipartite
(d) None of the previous
_____________________________________________________________________________________
(11) A tree with 50 vertices has
(a) 51 edges
(b) 49 edges
(c) 50 edges
(d) None of the previous
_____________________________________________________________________________________
(12) The dual of the Boolean expression 𝑥. 𝑧̅ + 𝑥. 0 + 𝑥̅ . 1 is
(a) (𝑥 + 𝑧̅). (𝑥 + 1). (𝑥̅ + 0)
(b) (𝑥̅ + 𝑧). (𝑥̅ + 1). (𝑥 + 0)
(c) (𝑥. 𝑧̅). (𝑥 + 0). (𝑥̅ + 1)
(d) None of the previous
_____________________________________________________________________________________
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Question II:
A. Without using truth tables prove that
(𝑝 ∧ 𝑞) → (𝑝 ∨ 𝑞)
is a tautology.
B. Prove by contradiction that if 𝑥 is any rational number and 𝑦 is any irrational number then 𝑥 − 𝑦 is an
irrational number.
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C. Using Mathematical Induction prove that
20 + 2. 21 + 3. 22 + ⋯ + 𝑛. 2𝑛−1 = (𝑛 − 1)2𝑛 + 1,
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𝑛 ≥ 1.
Question III:
A. Prove that the relation 𝑅 defined by:
𝑎𝑅𝑏 ⟺ 𝑎 − 𝑏 ∈ ℤ
is an equivalence relation on the set ℤ.
B. Let 𝑅 = {(𝑎, 𝑎), (𝑎, 𝑏), (𝑎, 𝑐), (𝑏, 𝑏), (𝑏, 𝑑), (𝑐, 𝑐), (𝑑, 𝑑)} be a relation on the set 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑}. Then draw
the diagraph representing the relation 𝑅. Is the relation 𝑅 antisymmetric? Justify your answer.
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C. Answer the following questions about the following graph 𝐺:
G
(i) Is the graph connected? Justify your answer.
(ii) Find 𝑑𝑒𝑔(𝑎), 𝑎𝑛𝑑 deg(𝑑). Find a path from 𝑏 𝑡𝑜 𝑒. What is its length?
(iii) Draw the subgraph of 𝐺 induced by the vertices 𝑎, 𝑏, 𝑐.
(iv) Draw a planar representation of the graph 𝐺 and find how many regions does it have? Justify your answer.
(v) Is the graph 𝐺 isomorphic to the graph below? Justify your answer.
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Question IV:
A. Answer the following questions about the tree below:
(i) Which vertex is the root? Which vertices are internal? Which vertices are leaves?
(ii) Which vertex is the parent of 𝑔? Which vertices are siblings of 𝑑?
(iii) Is the tree a 3-ary tree? Is it a full 3-ary tree? Justify your answer.
(iv) Find the level of the vertex 𝑐 and the vertex 𝑚. What is the height of the tree? Is the tree balanced? Justify
your answer.
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B. Draw a spanning tree of the graph below:
C. Draw a binary search tree for the words: mango, banana, peach, apple, coconut, pear, papaya (using
alphabetical order).
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Question V:
A. Find a Boolean expression for the function that has the values in the following table:
𝒙
𝒚
𝒛
𝑭(𝒙, 𝒚, 𝒛)
1
1
1
1
1
1
0
0
1
0
1
1
1
0
0
0
0
1
1
0
0
1
0
1
0
0
1
0
0
0
0
1
B. Find the sum of product expansion for the following Boolean function:
𝑓 (𝑥, 𝑦, 𝑧) = 𝑦𝑧̅ + 𝑥𝑦.
Find the value of 𝑓 (1,1,0).
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C. Construct a circuit from inverters, AND gates, and OR gates to produce the output
𝑥𝑦̅ + 𝑥̅ 𝑦̅.
D. Use K-maps to minimize the sum of product expansion
𝑥𝑦𝑧 + 𝑥𝑦𝑧̅ + 𝑥̅ 𝑦𝑧̅ + 𝑥̅ 𝑦𝑧 + 𝑥𝑦̅𝑧.
Good Luck 
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