Let's analyze the provided statements and classify each as either True or False:

1. (a) If $A = \{h, u, g, h\}$ and $B = \{h, u, g\}$, then $|A| = |B|$.

   • Explanation: The set $A = \{h, u, g, h\}$ includes the element "h" twice, but in sets, duplicate elements are not counted. Therefore, $A = \{h, u, g\}$, which is identical to set $B$. Since both sets have the same elements, their cardinalities $|A|$ and $|B|$ are equal.
   • Answer: True

2. (b) $(-1/5)^{-2} = 25$.

   • Explanation: The expression $(-1/5)^{-2}$ means taking the reciprocal of $(-1/5)^2$. First, squaring $-1/5$ gives $1/25$, and taking the reciprocal of $1/25$ results in 25.
   • Answer: True

3. (c) $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$.

   • Explanation: The equation $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ is generally false. The square root of a sum is not equal to the sum of the square roots unless both $a$ and $b$ are zero, but in general, this equality does not hold.
   • Answer: False

4. (d) The inequality $x^2 + 1 < 1$ has no solution.

   • Explanation: The inequality $x^2 + 1 < 1$ implies that $x^2 < 0$. However, since $x^2 \geq 0$ for all real numbers $x$, there is no real solution to this inequality.
   • Answer: True

Would you like further details on any of these solutions?

Related Questions:

1. How do you calculate the cardinality of a set?
2. What is the general rule for handling negative exponents?
3. When does the equation $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ hold true?
4. How do you solve inequalities involving squares of variables?
5. Can you explain the concept of sets and how duplicates are treated in set theory?

Tip: Always double-check if the given elements in a set are distinct, as sets do not count repeated elements.