Analysis 1 (MATS2101)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2013
1. Suppose that x, y. z
∈ ℕ
Prove that if x ≠ y then x + z≠ y + z.
2. Prove that 0.0=0, 2.2=4, 2.4=8. 1.2=2.1=2, 3.3=9
3. Prove (the following distributive law in
ℕ
) that for all x, y, z
∈
ℕ
, (x + y)z = xz + yz (Hint: choose
and fix arbitrary x. y
∈ ℕ
and define a suitable set S
xy
then use the induction axiom.)
4. Using only the commutative laws for addition and (or) multiplication in
ℕ
and question 3.. prove
(the following distributive law in
ℕ
) that for all
x.
y, z
∈ ℕ
, z(x
+
y)=
xz
+ yz.
5. Prove that for all x, y, z ∈ ℕ
. (xy)z =
x(yz)
6. Prove that for all m ∈ ℕ, m ≠ 0. m
0
= 1 and O
m
=0.
7. Wh a t i s 0
o
?
8 Prove that for all m. n ∈ ℕ. m
0
=0
0
= 0
9 Prove that if m. n, b are natural numbers such that m = n + b and b ≠ 0
then m≠ n and hence that if m < n then m ≠n.
10. Prove that if m. n ∈ ℕ then it is not possible that m < n and n < m.
11. Prove that if m. n∈ ℕ and nm = 0 then m = 0 or n = 0.
12. A strictly totally ordered set (S. <) is a set on which an order relation " <" is
defined such that for all x ∈ S. y ∈ S, only one of the following occurs:
x < y, x=y o r y < x. We w ri te x ≤ y if x< y o r x = y .xy ≥ y if y ≤ x
and x > y if y <x. An clement m ∈ S is called a minimum if m ≤ x for all
x ∈ S and an element M ∈ S is called a maximum if x ≤ M for all x∈ S. If (U,
<) is a strictly totally ordered set and S ⊂ U, then m ∈ U\S is called a lower
bound for S if m ≤ x for all x ∈ S and an element M ∈ U∖S is called an
upper bound for S if x ≤ M for all x ∈ S.
(i) Show that if. (S; <) is a strictly totally ordered set, then the
relation "≤"on S is transitive.
(ii) Show that x < y is the negation of y ≤ x, x ≤ y is the negation of y
< x and if x ≤ y an d y ≤ then x = y .
(iii) Show that if a strictly totally ordered set S has a maximum then it is
unique and if it has a minimum then it is unique.
UNIVERSITY OF BAMENDA
FACULTY OF SCIENCE
MATS2101 TUTORIALS
Department: Mathematics and Computer Science Course Instructor: Dr. Shu F. CHE.
January 19, 2014
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2
13.
Show that for m ∈ , m - 0 = m.
14.
Let m. n ∈ ℕ and ∆ denote the net difference function, Prove that ∆(m,
0)= (m. 0) and ∆ (0, n)(0 n) and hence that ∆Ω ∆ - ∆
15.
Prove that (ℤ,+, .) is a commutative ring, where " + " denotes the
addition and "." the multiplication defined on it.
16.
Prove each of the following statements about elements x, y of a ring
(R, +,.) where “denotes” addition and “.” Multiplication in the ring:
0.x = x.0 = 0, (-x).y = - (x.y) = x.(-y), (-1) = y(- 1) = - y (- x) = xy
17.
Prove that if x, y ∈ ℤ are such that x ≠ 0 and y ≠0 then xy ≠ 0.
18.
An integral domain is a ring (R,+,.) such that 0≠1 and for all x. y ∈ R
xy = 0 implies x = 0 or y = 0. Is (ℤ, +,.) an integral domain?
19. Show that if x, y, z ∈ ℤ are such that x < y and z < 0 then yz < xz.
20. Prove that multiplication in ℚ is well defined.
20. Prove that (ℚ, +, .) is an ordered field.
22. Prove that if
F
is a field and
x
∈
F,
then |x| =
max {
x,
-x}.
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