Solutions Manal For An Introduction to Metric Spaces, 1st Edition By Dhananjay Gopal, Aniruddha Deshmukh

Solutions Manal For An Introduction to Metric Spaces, 1st Edition By
Dhananjay Gopal, Aniruddha Deshmukh (All Chapters 1-7, 100%
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The Textbook This Document Fully Covers
An Introduction to Metric Spaces, 1st Edition By Dhananjay Gopal,
Aniruddha Deshmukh




• Publisher : Chapman and Hall/CRC
• Publication date : July 14, 2020
• Edition : 1st
• Language : English
• Print length : 302 pages
• ISBN-10 : 0367493489
• ISBN-13 : 978-0367493486

, Solutions Manual for An Introduction to Metric
Spaces, 1e Dhananjay Gopal, Aniruddha
Solutions Manual for An Introduction to Metric
Deshmukh


Spaces, 1e Dhananjay Gopal, Aniruddha
Deshmukh (All Chapters)
Chapter 1
Set Theory

Ex 1.1. Collection of lines with constant slope and changing intercepts (or, vice-versa) does the job.
That is,
F = {Lr |r ∈ R} ,
where, Lr = {(x, y) ∈ R2 |x + y = r} are lines with slope −1 and intercept r. Other similar sets can be
constructed which can be indexed by R.

Ex 1.2. A few subsethood relations are

N⊆Z⊆Q⊆R
N ⊆ Q+ ⊆ R+ ⊆ R
N ⊆ Q∗ ⊆ R∗ ⊆ R

Similar relations can be obtained by comparing elements.

Ex 1.3. The results for N are given. Other results can be constructed in a similar manner.

N∪Z=Z
N∩Z=N
N∪Q=Q
N∩Q=N
N ∪ Q+ = Q+
N ∩ Q+ = N
N ∪ Q− = {x ∈ Q|x < 0 or x ∈ N}
N ∩ Q− = ∅
N ∪ Q∗ = Q∗
N ∩ Q∗ = N
N ∪ R + = R+
N ∩ R+ = N
N ∪ R− = {x ∈ R|x < 0 or x ∈ N}
N ∩ R− = ∅
N ∪ R∗ = R∗
N ∩ R∗ = N

A few observations that can be made are as follows. The reader can make as many observations as they
want to.

1. A ⊂ B implies A ∪ B = B and A ∩Manual
Solutions for An Introduction to Metric
B = A.
Spaces, 1e Dhananjay Gopal, Aniruddha
Deshmukh
2. Some of the (newly formed) sets are merely a result of union/intersection of known sets.

,