Download Question IV: a)Let be a relation defined on defined by . i)Prove that

King Saud University
MATH 131
Department of Mathematics
Final Exam
1st Semester 1435-1436H
Duration: 180 Minutes
‫ ال يسمح باالجابة بالقلم الرصاص‬:‫مالحظة‬
Student’s Name
Student’s ID
Question
Number
II
I
III
Group Number
IV
V
VI
Lecturer’s Name
VII
VIII IX
XTotal
Mark
Question
Number
1
2
3
4
5
6
7
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Answer
Question I:
Choose the correct answer, then fill in the table above:
(1) ~[∀𝒙 ∈ ℝ, 𝑰𝒇 𝒙 = 𝟏 𝒐𝒓 𝒙 = −𝟏, 𝒕𝒉𝒆𝒏 |𝒙| = 𝟏] ≡
(a) ∀𝒙 ∈ ℝ, 𝑰𝒇 𝒙 ≠ 𝟏 𝒂𝒏𝒅 𝒙 ≠ −𝟏, 𝒕𝒉𝒆𝒏 |𝒙| = 𝟏.
(b) ∃𝒙 ∈ ℝ, 𝑰𝒇 𝒙 ≠ 𝟏 𝒂𝒏𝒅 𝒙 = −𝟏, 𝒕𝒉𝒆𝒏 |𝒙| ≠ 𝟏.
(c) ∃𝒙 ∈ ℝ, 𝒙 = 𝟏 𝒂𝒏𝒅 𝒙 = −𝟏 𝒂𝒏𝒅 |𝒙| ≠ 𝟏.
(c) Non of the previous.
(2) The contrapositive of :" 𝒏 = 𝟐 𝒐𝒓 𝒏 𝒊𝒔 𝒐𝒅𝒅, 𝒘𝒉𝒆𝒏𝒆𝒗𝒆𝒓 𝒏 𝒊𝒔 𝒑𝒓𝒊𝒎𝒆. " Is:
(a) 𝑰𝒇 𝒏 = 𝟐 𝒐𝒓 𝒏 𝒊𝒔 𝒐𝒅𝒅, 𝒕𝒉𝒆𝒏 𝒏 𝒊𝒔 𝒑𝒓𝒊𝒎𝒆.
(b) 𝑰𝒇 𝒏 ≠ 𝟐 𝒂𝒏𝒅 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏 𝒕𝒉𝒆𝒏 𝒏 𝒊𝒔 𝒏𝒐𝒕 𝒑𝒓𝒊𝒎𝒆.
(c) 𝑰𝒇 𝒏 𝒊𝒔 𝒏𝒐𝒕 𝒑𝒓𝒊𝒎𝒆 𝒕𝒉𝒆𝒏 𝒏 ≠ 𝟐 𝒂𝒏𝒅 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏.
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10
Total
(d) none of the previous.
(3) The set ℚ+ ∩ (𝟎, 𝟏) is
(a) a finite set.
(b) a denumerable set. (c) not countable. (d) non of the previous.
(4) Let 𝑹 = {(𝒂, 𝒃): 𝒂 ≠ 𝒃, 𝒂, 𝒃 ∈ ℕ} be a relation on ℕ, then R is
(a) reflexive and symmetric.
(b) transitive and not symmetric.
(c) not reflexive and symmetric.
(d) none of the previous.
(5) Let R be a relation on ℤ defined by 𝒎𝑹𝒏 if and only if 𝒎 < 2𝒏, then R is
(a) a linear ordering.
(b) an equivalence relation.
(c) a partial ordering
(d)none of the previous.
(6) Let 𝒇(𝒙) = |𝒙|, 𝒙 ∈ ℝ , then:
(a) 𝒇 is one to one.(b) 𝒇 is onto.(c) 𝒇 is not one to one and not onto.
bijection.
(d) 𝒇 is a
(7) If 𝒇: 𝑨 → 𝑩, 𝒈: 𝑩 → 𝑪 and 𝒈𝝄𝒇: 𝑨 → 𝑪 is onto, then
(a) 𝒇 is onto.
(b) 𝒈 is onto.
(c) 𝒇 and 𝒈 are both onto.
d) non of the previous.
(8) If 𝒇(𝒙) = 𝒆𝒙 , 𝒙 ∈ (𝟎, ∞) and 𝒈(𝒙) = |𝒙|, 𝒙 ∈ [−𝟏𝟎, 𝟏𝟎], then 𝒇 ∪ 𝒈 is
(a) a one to one function.
(b) an onto function.
(c) not a function.
d) non of the previous.
(9) If 𝒇(𝒙) = 𝒙𝟐 + 𝟏, then 𝒇−𝟏 {[𝟓, 𝟏𝟎]} =
(a) [√𝟓, √𝟏𝟎]
(b) [−𝟑, −𝟐] ∪ [𝟐, 𝟑]
(10)𝐋𝐞𝐭 𝐀 ℝ 𝐭𝐡𝐞𝐧 𝐭𝐡𝐞 𝐬𝐞𝐭 𝐁 {
⊆ ,
= 𝐱𝛜ℝ:
(c) [4,9]
(𝐱) = 𝟏 }
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(d) non of the previous.
Question II:
Mark true (√) or false (×) in front of the following statements. Give reasons.
(𝟏)𝐅𝐨𝐫 𝐚𝐥𝐥 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞 𝐢𝐧𝐭𝐞𝐠𝐞𝐫𝐬,
𝒙𝟐 + 𝒙 + 𝟒𝟏 𝒊𝒔 𝒂 𝒑𝒓𝒊𝒎𝒆. [
(𝟐)𝐈𝐟 𝐴 ⊆ 𝐵, 𝐭𝐡𝐞𝐧 𝐵 − 𝐴 = 𝜙. [
]
]
(𝟑)𝐋𝐞𝐭 𝐀 𝐚𝐧𝐝 𝐁 𝐛𝐞 𝐬𝐞𝐭𝐬, 𝐢𝐟 𝐀 ⊆ 𝐁, 𝐭𝐡𝐞𝐧 𝐁 𝐢𝐬 𝐢𝐧𝐟𝐢𝐧𝐢𝐭𝐞 𝐢𝐟 𝐀 𝐢𝐬 𝐢𝐧𝐟𝐢𝐧𝐢𝐭𝐞. [
(𝟒)ℤ 𝒊𝒔 𝒂 𝒅𝒆𝒏𝒖𝒎𝒆𝒓𝒂𝒃𝒍𝒆 𝒔𝒆𝒕. [
]
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]
Question III:
𝒂)𝐔𝐬𝐞 𝐦𝐚𝐭𝐡𝐞𝐦𝐚𝐭𝐢𝐜𝐚𝐥 𝐢𝐧𝐝𝐮𝐜𝐭𝐢𝐨𝐧𝐭𝐨 𝐬𝐡𝐨𝐰 𝐭𝐡𝐚𝐭
𝟏 𝟐
𝒏
𝟏
+ +⋯+
=𝟏−
.
(𝒏 + 𝟏)!
(𝒏 + 𝟏)!
𝟐! 𝟑!
b)Prove that for all integers a,b and c,
if a divides b-1 and a divides c-1, then a divides bc-1.
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c) 𝐏𝐫𝐨𝐯𝐞 𝐭𝐡𝐚𝐭: √𝟐 is an irrational number.
Question IV:
a)Let 𝑹 be a relation defined on ℤ defined by
𝒙𝑹𝒚 ↔ |𝒙 − 𝟑| = |𝒚 − 𝟑|.
i)Prove that 𝑹 is an equivalence relation.
ii)Find [1] and [3].
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b)If 𝑨, 𝑩, 𝑪 and 𝑫 are sets. Prove that if 𝑪 ⊆ 𝑨, 𝑫 ⊆ 𝑩 and 𝑨 and 𝑩 are disjoint,
then 𝑪 and 𝑫 are disjoint.
c)Show that 𝑨 × (𝑩 ∩ 𝑪) = (𝑨 × 𝑩) ∩ (𝑨 × 𝑪).
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Question V:
𝒙−𝟐
, 𝒙 ≠ −𝟒
(a) Let 𝒇(𝒙) = {𝒙+𝟒
then
𝟏
𝒙 = −𝟒
(i)
Prove that 𝒇 is one to one and onto ℝ.
(ii) Find 𝒇−𝟏 .
(b) Let 𝒇: 𝑨 → 𝑩, 𝑫 ⊆ 𝑨, and 𝑬 ⊆ 𝑩. Prove that 𝑨 − 𝒇−𝟏 (𝑬) ⊆ 𝒇−𝟏 (𝑩 − 𝑬).
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Bonus Question
Prove that equivalence of sets ≈ is an equivalence relation.
Good luck
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