Analysis 1 (MATS2101)
BSc, Physics - PHYS
Semester: First Semester
Level: 200
Year: 2016
REPUBLIC OF CAMEROON REPUBLIQUE DU
-
CAMEROUN
Peace-Worlc-Fatherland Paix - Travaille - Patrie
MINISTRY OF HIGHER MINISTERE: DE L'ENSEIGNEMENT
EDUCATION SUPERIEURE
THE UNIVERSITY OF BAMENDA
UNIVERSITE DE BAMENDA
FACULTY OF SCIENCE
P.O Box 39 BAMBILI
TEL: (+23) 22 31 63 50
FACULTE DE SCTENCE
TUTORIAL SHEET 2 2015/2016
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
COURSE CODE: M.A.TS2101 SHU F. CHE
Answer all questions.
1. Prove the following equalities for the given integers: (i) 7 - 7 = 0
(ii)
7 - 4 = 3
(iii)
4 - 7 = -3.
2. Show that for n ∈ ℕ, ∆ (n, 0) = (n, 0) and ∆ (0, n). = (0, n).
3. Prove that for all n ∈ ℕ, n > 0, [n, 0] ≠ [0, n]
4. Prove that subtraction in ℤ is not commutative while addition in ℤ is
commutative.
5. Prove that subtraction in
ℤ
is associative.
6. Prove that subtraction in
ℤ
is
well defined.
7. Prove that
there exists
a bijective function
f : ℕ → ℤ
8. Prove
that [m, n] =
[∆(, )]
for all m, n
∈
ℤ
.
9. Let a.
b
∈
ℤ
.
. Find
x
∈
ℤ
such
that
x +
a =
b.
10. Lot a,
b
∈
ℤ
. Show that - (a +
b) = (
-
a)
+ (-b).
11. Show that If (G,∗) is a group, and e is the identity
in (G,∗) then its inverse
is e.
12. Prove each of the
following statements for a, b, c
∈
ℤ
:
(i) a – b = 0 ⟺ a = b
(ii) a ≤ b ⟺b -a
∈
ℕ
.
(iii)
if a <
b
then a +
c
<
b + c
13. Prove that (Z, +,∗) with the addition and multiplication defined as in the lecture is a ring.
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