Note to the Reader

We shall use standard set-theoretic notations such as

∪, ∩, ⊆, ⊂, ∈

to denote ‘union’, ‘intersection’, ‘subset of’, ‘proper subset of’, ‘belong(s) to’, respectively. For sets S1 and S2, the set {x ∈ S1 : x ∉ S2} is denoted by S1 \ S2. If f is a function with domain S1 and codomain S2, then we use the notation f : S1 → S2, and it is also called a ‘map’ from S1 to S2.

Also, we use the following standard notations and symbols:

ℕ

:

set of all positive integers

ℤ

:

set of all integers

ℝ

:

set of all real numbers

ℂ

:

set of all complex numbers

: =

:

is defined by

∀

:

for all

∃

:

there exists or there exist

⇒

:

implies or imply

⇔

:

if and only if

↦

:

maps to

To mark the end of a proof (of a lemma, proposition, theorem, or corollary), we use the symbol ▮, while the symbol ♢, is used to mark the ends of definitions, remarks, examples, and exercises.

Numbering of definitions, results (lemmas, theorems, propositions, corollaries), remarks, examples, and exercises are done consecutively using three digits p.q.r, where p and q denote the chapter number and section number, respectively, and r denotes its actual occurrence.