King Saud University
Science and Medical Studies Section for
girls
College of Science
Department of Mathematics
Math 151
First Term 1433-34H
Final Exam
3 Hours
Name:
Student No.:
Section No.:
Staff member name:
Question
No
Mark
1
2
3
4
Total
Question 1
Choose the correct answer and write it in the following table:
Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Answer
1) The compound propositions ((𝒑 → 𝒓) ∨ (𝒓 → 𝒒)) ∧ 𝒓 and the proposition r
are logically equivalent
a) true
b) false
2) The compound proposition [(𝒑 ∧ 𝒒) ∨ (𝒑 → 𝒓) ∨ (𝒒 → 𝒓)] → 𝒓 is
a) a tautology
b) a contradiction
c) a
contingency
3) The proposition “ ∀ 𝒙, 𝒙𝟐 < 4 𝑎𝑛𝑑 𝑥 = 1, 𝑡ℎ𝑒𝑛 𝒙𝟐 + 𝟏 = 𝟐 ” is
a) true
b) false
4) Let 𝑹 be the equivalent relation “𝒂𝑹𝒃 ⇔ 𝒂 ≡ 𝒃(𝒎𝒐𝒅 𝟕)”. Then
a) [𝟏𝟒] = [𝟕].
b) [𝟓] = {… − 𝟐𝟓, −𝟓, 𝟎, 𝟓, 𝟐𝟓, … ).
c) Both (a) and (b) are true.
c) None of the previous.
5) Which of the next relations is a partial order relation on 𝒁+ ?
(a) 𝒂𝑹𝒃 ⇔ 𝒂 ≡ 𝒃(𝒎𝒐𝒅 𝟐𝟎).
(b) 𝒂𝑹𝒃 ⇔ 𝒂 |𝒃 , ( 𝒂 𝒅𝒊𝒗𝒊𝒅𝒆𝒔 𝒃). .
(c) 𝒂𝑹𝒃 ⇔ 𝒂 > 𝑏.
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6) The relation represented by the following directed graph is
a)reflexive
d) transitive.
b) symmetric.
c) antisymmetric.
e) non of the previous.
7) The diagonal relation 𝚫 is a partial order relation and an equivalence
relation at the same time.
a) True
b) False
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
̅̅̅̅̅̅̅̅̅̅
̅) + 𝒚
̅) is a Boolean function, then 𝒇(𝒙, 𝒚, 𝒛) is
8) If 𝒇(𝒙, 𝒚, 𝒛) = ((𝒙
+𝒚
equivalent to
a) 𝟎
d) None of the previous.
̅
b) 𝒙𝒚
̅+𝒚
̅
c) 𝒙𝒚
9) If 𝒇 𝒊𝒔 as in (8), then the dual of 𝒇 is
̅̅̅̅̅̅̅̅̅̅
̅̅̅̅̅̅
̅) + 𝒚
̅
̅)𝒚
̅)
a) (𝒙
+𝒚
b) ̅̅̅̅̅̅̅̅̅̅̅
((𝒙𝒚
̅𝒚 + 𝒚
𝒙
c)
d) None of the previous.
10) Assume that 𝒙, 𝒚, and 𝒛 represent Boolean variables, then ((𝒙 + 𝒛) +
𝒚)𝒙 + ̅
𝒙 = 𝟏.
a) true
b) false
11) There is a simple graph with six vertices, whose degree sequence is 12, 2, 2,
3, 5, 4.
a) true
b) false
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12) The graph D is bipartite
a) True
b) False
13) The graph 𝑲𝒏 has
a) 𝒏 vertices and 𝒏𝟐 edges.
c) 𝒏 vertices and 𝒏(𝒏 − 𝟏) edges.
b) 𝟐𝒏 vertices and 𝒏𝟐 edges.
14) A subgraph of a simple graph is simple.
a) true
b) false
15) If G is a planar connected graph with 20 vertices, each of degree 3, then G
has 12 regions.
a) true
b) false
16) There is a connected simple planar graph with 10 vertices and 16 edges
and 8 regions.
a) true
b) false
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Question 2
Prove the following statements:
1) ∼ (𝒑 ↔ 𝒒) ≡ 𝒑 ↔∼ 𝒒 .
2) √𝟐 ∉ ℚ.
̅ + 𝒃𝒄 = 𝟎 then 𝒂
̅ (𝒃 +
3) Let 𝑺 be a Boolean algebra, and 𝒂, 𝒃 ∈ 𝑺 such that 𝒂
𝒄) = 𝟏.
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4) Let 𝑿 ≠ ∅. Define on ℘(𝑿) the next operations:
𝑨+𝑩=𝑨∪𝑩
𝑨. 𝑩 = 𝑨 ∩ 𝑩
𝑨′ = 𝑿 − 𝑨
𝟎=∅
𝟏 = 𝑿.
′
Then, (𝑿, +, . , , 𝟎, 𝟏) is a Boolean algebra.
5) Let 𝑮 be a simple connected planar graph with 5 vertices then |E| ≤ 𝟗.
6) This graph is planar.
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Question 3
I)
Let 𝒂𝟎 = 𝟏, 𝒂𝟏 = 𝟏, 𝒂𝒏 = 𝒂𝒏−𝟏 + 𝒂𝒏−𝟐 for all 𝒏 ≥ 𝟐. Prove by using
the second principle of mathematical induction that
𝒂𝒏 ≤ 𝟐𝒏 for all 𝒏 ≥ 𝟎.
II)
Are the two graphs isomorphic?
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III)
a) Draw the graph 𝑲𝟑,𝟒
b) Prove that 𝑲𝟑,𝟒 is not a planar graph.
IV) State the Euler formula for a connected simple graph.
V) If G is a connected graph with 12 regions and 20 edges, then G has
________ vertices.
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Question 4
Use Karnaugh maps (K-maps) to simplify the following Boolean function
̅ +𝒙
̅𝒚
̅𝒛 + 𝒚𝒛 ,
𝒇(𝒙, 𝒚, 𝒛) = 𝒙𝒚
and then draw the minimal “and” and “or” circuit. (‫)شبكة عطف و فصل أصغرية‬
Good Luck
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