Regular expression properties (theory) [closed] - regex

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I'm reading Aho and Ullman's book The Theory of Parsing, Translation, and Compiling. In the section that introduces regular expressions in chapter 2, there is a list of properties of regular expressions. I do not understand properties 2 and 8. Here is the list of properties:
(1) 𝛼 + 𝛽 = 𝛽 + 𝛼
(2) ∅* = 𝑒
(3) 𝛼 + (𝛽 + 𝛾) = (𝛼 + 𝛽) + 𝛾
(4) 𝛼(𝛽𝛾) = (𝛼𝛽)𝛾
(5) 𝛼(𝛽 + 𝛾) = 𝛼𝛽 + 𝛼𝛾
(6) (𝛼 + 𝛽)𝛾 = 𝛼𝛾 + 𝛽𝛾
(7) 𝛼𝑒 = 𝑒𝛼 = 𝛼
(8) ∅𝛼 = 𝛼∅ = ∅
(9) 𝛼* = 𝛼 + 𝛼*
(10) (𝛼* )* = 𝛼*
(11) 𝛼 + 𝛼 = 𝛼
(12) 𝛼 + ∅ = 𝛼
where ∅ is the regular expression denoting the regular set ∅, 𝛼, 𝛽, 𝛾 are arbitrary regular expressions, and 𝑒 is the empty string.
How are properties (2) and (8) justified?
Edit: To explain the notation of +, *, etc, here are some definitions given in the book (quoted):
DEFINITION Let 𝚺 be a finite alphabet. We define a regular set over
𝚺 recursively in the following manner:
(1) ∅ (the empty set) is a regular set over 𝚺.
(2) {𝑒} is a regular set over 𝚺.
(3) {𝑎} is a regular set over 𝚺 for all 𝛼 in 𝚺.
(4) If 𝑃 and 𝑄 are regular sets over 𝚺, then so are
(a) 𝑃 ∪ 𝑄.
(b) 𝑃𝑄.
(c) 𝑃*.
(5) Nothing else is a regular set.
Thus a subset of 𝚺* is regular if and only if it is ∅, {𝑒}, or {𝑎},
for some 𝑎 in 𝚺, or can be obtained from these by a finite number of
applications of the operations union, concatenation, and closure.
.
DEFINITION Regular expressions over 𝚺 and the regular expressions
they denote are defined recursively, as follows:
(1) ∅ is a regular expression denoting the regular set ∅.
(2) 𝑒 is a regular expression denoting the regular set {𝑒}.
(3) 𝑎 in 𝚺 is a regular expression denoting the regular set {𝑎}.
(4) If 𝑝 and 𝑞 are regular expressions denoting the regular sets 𝑃
and 𝑄, respectively, then
(a) (𝑝+𝑞) is a regular expression denoting 𝑃 ∪ 𝑄.
(b) (𝑝𝑞) is a regular expression denoting 𝑃𝑄.
(c) (𝑝)* is a regular expression denoting 𝑃*.
(5) Nothing else is a regular expression.

Answer 1

My guess is that 2 & 8 properties might be just a simple math:
Property 2
∅ is an empty set, then ∅* = 𝑒 is true, ∅+ = 𝑒 is also true, ∅{Infinity} = 𝑒 is also true, since e is an empty string.
A regular expression is a string, thus an empty regular expression repeating any number of times or with any operation, is still an empty regular expression, which again equals to an empty string in the right side.
Reference:
Why is the Kleene star of a null set is an empty string?
Property 8
∅𝛼 = 𝛼∅ = ∅ is true, and so is ∅𝛼∅𝛼∅𝛼 = 𝛼∅𝛼∅𝛼∅ = ∅, because an empty set combined with anything would result an empty set.
Reference:
Regular expressions with empty set/empty string
What is the difference between language of empty string and empty set language?
How can concatenating empty sets (languages) result in a set containing empty string?