Analysis Total Found Anser

Section-b
11ans) pseudometric space is a generalization of a metric space in which the distance between
two distinct points can be zero. In the same way as every normed space is a metric space,
every seminormed space is a pseudometric space. Because of this analogy the term semimetric
space (which has a different meaning in topology) is sometimes used as a synonym, especially
in functional analysis.
When a topology is generated using a family of pseudometrics, the space is called a gauge space.
The concept of a quasi-partial-metric space was introduced by Karapınar et al. [17]. He studied
some fixed point theorems on these spaces whereas Shatanawi and Pitea [18] studied some
coupled fixed point theorems on quasi-partial-metric spaces.
The aim of this paper is to introduce the concept of quasi-partial b-metric spaces which is a
generalization of the concept of quasi-partial-metric spaces. The fixed point results are proved in
setting of such spaces and some examples are given to verify the effectiveness of the main
results.
A quasi-partial metric on a non-empty set X is a function q:X×X→R+q:X×X→R+,

satisfying
(QPM1):
If q(x,x)=q(x,y)=q(y,y)q(x,x)=q(x,y)=q(y,y), then x=yx=y.
(QPM2):
q(x,x)≤q(x,y)q(x,x)≤q(x,y).
(QPM3):
q(x,x)≤q(y,x)q(x,x)≤q(y,x).
pseudometric space is a generalization of a metric space in which the distance between two
distinct points can be zero. In the same way as every normed space is a metric space,
every seminormed space is a pseudometric space. Because of this analogy the term semimetric
space (which has a different meaning in topology) is sometimes used as a synonym, especially
in functional analysis.
When a topology is generated using a family of pseudometrics, the space is called a gauge space.
18ans)
12ans) In mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is
named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in
the Journal de Mathématiques Pures et Appliquées in 1862
Let
(1)
where is independent of . Then if and
(2)
it follows that
To do this we show that the Cauchy criterion holds.
Assign ϵ<0ϵ<0.
Then by definition of uniform convergence:
∃N∈N:∀x∈D:∀n≥N:|bn(x)|<ϵ2∃N∈N:∀x∈D:∀n≥N:|bn(x)|<ϵ2
Let x∈Dx∈D and n>m≥Nn>m≥N.
Then:
∑k=m+1n|bk(x)−bk+1(x)|∑k=⁡m+ = ∑k=m+1n(bk(x)−bk+1(x))∑k=⁡m+
1n|bk(x)−bk+1(x)| = 1n(bk(x)−bk+1(x))
= bm+1(x)−bn+1(x)bm+1(x)−bn+1(x
= )
= |bm+1(x)−bn+1(x)||bm+1(x)−bn+1
= (x)|
≤ |bm+1(x)+bn+1(x)||bm+1(x)+bn+1
≤ (x)|
< ϵ2+ϵ2ϵ2+ϵ2
<
= ϵ
=
The Union and Intersection of Collections of

15ans)
Closed Sets
Recall from The Union and Intersection of Collections of Open Sets page that if F is an arbitrary collection
of open sets then ⋃A∈FA is an open set, and if F={A1,A2,...,An} is a finite collection of open sets
then ⋂i=1nAi is an open set. We will now prove two analogous theorems regarding the union and
intersection of collections of closed sets.
Theorem 1: If F={A1,A2,...,An} is a finite collection of closed sets then ⋃i=1nAi is a closed set.
• Proof: Let F={A1,A2,...,An} be a finite collection of closed sets and let:
(1)
S=⋃i=1nAi
• By applying the generalized De Morgan's Law, we see that the complement Sc is:
(2)
Sc=(⋃i=1nAi)c=⋂i=1nAci
• For each Ai for i∈{1,2,...,n} we have that Ai is closed, so Aci is open. The intersection of a finite
collection of open sets is open, so Sc is open and hence (Sc)c=S is closed. Therefore ⋃i=1nAi is
closed. ■
Theorem 2: If F is an arbitrary collection of closed sets then ⋂A∈FA is a closed set.
• Proof: Let F be any arbitrary collection of closed sets and let:
(3)
S=⋂A∈FA
• By applying the generalized De Morgan's Law, we see that the complement Sc is:
(4)
Sc=(⋂A∈FA)c=⋃A∈FAc
• For all A∈F we have that A is closed, so Ac is open. The union of an arbitrary collection of open
sets is open, so Sc is open. Therefore (Sc)c=S is closed. ■
16ans)∣z1+z2∣2=∣z1∣2+∣z2∣2
⇒(z1+z2)(z1ˉ+z2ˉ)=∣z1∣2+∣z2∣2
⇒z1z2ˉ+z1ˉz2=0
⇒z1z2ˉ+z1z2ˉˉ=0
Therefore, z1z2ˉ is purely imaginary
⇒z2z1+z2ˉz1ˉ=0
also z2z1 is purely imaginary.

⇒arg(z2z1)=2π
and hence, O,z1,z2 are vertices of right triangle.
17ans)
suppose √8 = a/b with integers a, b
and gcd(a,b) = 1 (meaning the ratio is simplified)
then 8 = a²/b²
and 8b² = a²
this implies 8 divides a² which also means 8 divides a.
so there exists a p within the integers such that:

a = 8p
and thus,
√8 = 8p/b
which implies
8 = 64p²/b²
which is:
1/8 = p²/b²
or:
b²/p² = 8
which implies
b² = 8p²
which implies 8 divides b² which means 8 divides b.
8 divides a, and 8 divides b, which is a contradiction because gcd (a, b) = 1

therefore, the square root of 8 is irrational.
Let us assume to the contrary that √3 is a rational number.

It can be expressed in the form of p/q
where p and q are co-primes and q≠ 0.
⇒ √3 = p/q
⇒ 3 = p2/q2 (Squaring on both the sides)
⇒ 3q2 = p2………………………………..(1)
It means that 3 divides p2 and also 3 divides p because each factor should appear two times for the
square to exist.
So we have p = 3r
where r is some integer.
⇒ p2 = 9r2………………………………..(2)
from equation (1) and (2)
⇒ 3q2 = 9r2
⇒ q2 = 3r2
Where q2 is multiply of 3 and also q is multiple of 3.
Then p, q have a common factor of 3. This runs contrary to their being co-primes. Consequently, p /
q is not a rational number. This demonstrates that √3 is an irrational number.
Section-c
21ans)
20ans)
22ans)

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Section-b

A quasi-partial metric on a non-empty set X is a function q:X×X→R+q:X×X→R+,

where is independent of . Then if and

The Union and Intersection of Collections of

• Proof: Let F={A1,A2,...,An} be a finite collection of closed sets and let:

Theorem 2: If F is an arbitrary collection of closed sets then ⋂A∈FA is a closed set.

• Proof: Let F be any arbitrary collection of closed sets and let:

also z2z1 is purely imaginary.

this implies 8 divides a² which also means 8 divides a.

so there exists a p within the integers such that:

8 divides a, and 8 divides b, which is a contradiction because gcd (a, b) = 1

Let us assume to the contrary that √3 is a rational number.

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