Here are the solutions to the questions:

a) $\{x \mid x \text{ is a real number such that } x^2 = 1\}$

• The solutions to $x^2 = 1$ are $x = 1$ and $x = -1$.
• Answer: $\{1, -1\}$

b) $\{x \mid x \text{ is a positive integer less than 12}\}$

• Positive integers less than 12 are $1, 2, 3, \dots, 11$.
• Answer: $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$

c) $\{x \mid x \text{ is the square of an integer and } x < 100\}$

• Squares of integers less than 100 are $1^2, 2^2, 3^2, \dots, 9^2$.
• Calculations: $1, 4, 9, 16, 25, 36, 49, 64, 81$.
• Answer: $\{1, 4, 9, 16, 25, 36, 49, 64, 81\}$

d) $\{x \mid x \text{ is an integer such that } x^2 = 2\}$

• The equation $x^2 = 2$ has solutions $x = \pm\sqrt{2}$, but $\pm\sqrt{2}$ are not integers.
• Answer: $\{\}$ (empty set)

Let me know if you'd like further clarifications or examples!

Expanding Questions:

1. How do we determine the roots of an equation like $x^2 = k$ when $k$ is a non-integer?
2. What are the properties of a set containing squares of integers?
3. Can you find examples of sets with no solutions and explain why?
4. How do we represent solutions graphically for these types of problems?
5. What happens if we extend these conditions to non-real numbers?

Tip: Always check the domain of the problem (real, integer, etc.) before finding solutions.