Let's analyze the statements one by one and determine whether they are true or false:

1. $\{ x \mid x \text{ is a letter in the name, "Peter"} \} = \{ x \mid x \text{ is a letter in the name, "John"} \}$
   False. The letters in "Peter" are {P, e, t, r}, and the letters in "John" are {J, o, h, n}. These two sets are not equal.

2. $\{ x \mid x \text{ is a digit in the number, "1,111,111"} \} = \{ x \mid x \text{ is a digit in the number, "11"} \}$
   True. In both cases, the only unique digit is "1".

3. $\{ x \mid x \text{ is a factor of 4} \} \sim \{ x \mid x \text{ is a factor of 10} \}$
   False. The factors of 4 are {1, 2, 4}, and the factors of 10 are {1, 2, 5, 10}. The sets are not equal.

4. $\{ x \mid x \text{ is a multiple of 2} \} = \{ x \mid x \text{ is an even number} \}$
   True. Multiples of 2 are all even numbers.

5. $\{ x \mid x \text{ is a letter in the word, "safe"} \} \subseteq \{ x \mid x \text{ is a letter in the word, "fairness"} \}$
   True. The letters in "safe" are {s, a, f, e}, and all these letters appear in "fairness" ({f, a, i, r, n, e, s}).

6. $\{1, 2, 3, 4, 5, 6\} \subseteq \{6, 5, 4, 3, 2, 1\}$
   True. The two sets are identical, meaning the subset relation holds.

7. $W \subseteq \mathbb{N}$
   This depends on the definition of $W$. If $W$ represents the set of whole numbers and $\mathbb{N}$ represents natural numbers, this is true as whole numbers include natural numbers.

8. $N \subseteq P$
   This cannot be determined without knowing the sets $N$ and $P$, but based on common set notation, we assume this is false unless stated otherwise.

9. $\{ x \mid x \text{ is a factor of 12} \} \subseteq \{ x \mid x \text{ is a multiple of 2} \}$
   True. The factors of 12 are {1, 2, 3, 4, 6, 12}, and all numbers except 1 are multiples of 2.

10. $\{ \} \subseteq \emptyset$
    True. The empty set is always a subset of any set, including itself.

Summary of Answers:

1. F
2. T
3. F
4. T
5. T
6. T
7. T
8. F
9. T
10. T

Would you like to ask further questions or dive into more details?

Here are five related questions to think about:

1. What is the difference between a subset and a proper subset?
2. How are factors and multiples different in terms of set relations?
3. Can a set be equal to its complement under any circumstances?
4. What properties do empty sets have in set theory?
5. How does equality between sets differ from subset relationships?

Tip: Always remember that the empty set is a subset of every set, but it's only equal to itself.