Example 1.8 1 What is the power set of the $$ \emptyset $$ ? What is the power set of $$ \{\emptyset\} $$ 2 Determine whether these statements are true or false. a. $$ 0 \in \emptyset $$ b. $$ \emptyset \in\{0\} $$ c. $$ \{0\} \in\{0\} $$ d. $$ 0 \subset \emptyset $$ e. $$ \{\emptyset\} \in\{\emptyset\} $$ f. $$ \{0\} \subseteq\{0 $$.

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We are given two parts to solve:

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Part 1. Find the power sets.

(a) Find the power set of ∅.

Recall that the power set of any set S, denoted ℘(S), is the collection of all subsets of S. Since the empty set ∅ has exactly one subset—namely, itself—we have:
  ℘(∅) = {∅}.

(b) Find the power set of {∅}.

Now consider S = {∅}. Its subsets are:
  1. The empty subset: ∅.
  2. The entire set: {∅}.

Thus:
  ℘({∅}) = {∅, {∅}}.

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Part 2. Determine whether the following statements are true or false.

Before we begin, note that in these problems the symbol “0” is used as an object distinct from the empty set ∅. Also, remember that:
• For any set A, “A ∈ B” means “A is an ELEMENT of B.”
• “A ⊆ B” means “A is a SUBSET of B” (possibly equal).
• “A ⊂ B” is sometimes used to denote “A is a PROPER subset of B” (i.e. A ⊆ B but A ≠ B).

Now let’s analyze each statement:

(a) 0 ∈ ∅ 
  Since ∅ has no elements at all, 0 is not an element of ∅.
  Answer: False.

(b) ∅ ∈ {0}
  The set {0} has exactly one element, namely the object 0. The empty set ∅ is not the same as 0.
  Answer: False.

(c) {0} ∈ {0}
  The only element of {0} is 0; the set {0} is not an element of itself.
  Answer: False.

(d) 0 ⊂ ∅
  Here “⊂” is taken to mean a proper subset. First, note that for an object to be a subset, it must be a set. Since 0 is given as an element (and not necessarily the empty set), it is not appropriate to consider it as a subset of anything. Even if one mistakenly thought that 0 might represent ∅, the empty set is not a proper subset of itself.
  Answer: False.

(e) {∅} ∈ {∅}
  The only element of {∅} is ∅; the set {∅} is not equal to ∅.
  Answer: False.

(f) {0} ⊆ {0}
  Any set is always a subset of itself.
  Answer: True.

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Summary of Answers:

1. ℘(∅) = {∅}  and  ℘({∅}) = {∅, {∅}}

2. a. False  
  b. False  
  c. False  
  d. False  
  e. False  
  f. True
**Power Sets:** - The power set of the empty set (∅) is {∅}. - The power set of {∅} is {∅, {∅}}. **Statement Evaluations:** - a. 0 ∈ ∅: False - b. ∅ ∈ {0}: False - c. {0} ∈ {0}: False - d. 0 ⊂ ∅: False - e. {∅} ∈ {∅}: False - f. {0} ⊆ {0}: True

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