Mathematics & Numerical Analysis

Analysis of complex modulus properties:

• Incorrect: The quotient property is an exact equality: $|z_1 / z_2| = |z_1| / |z_2|$.
• Incorrect: Denies the true equality.
• Correct: For any two complex numbers $z_1, z_2 \in \mathbb{C}$, the modulus of their product is strictly equal to the product of their individual moduli: $|z_1 z_2| = |z_1| |z_2|$.
• Incorrect: The strict inequality is false; equality always holds for multiplication.

Step-by-step conversion to polar form:
Given $z = x + iy = -4 + i 4\sqrt{3}$.

• Modulus $r$:
  $$r = \sqrt{x^2 + y^2} = \sqrt{(-4)^2 + (4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8$$
• Argument $\theta$:
  $$\cos \theta = \frac{x}{r} = \frac{-4}{8} = -\frac{1}{2}$$
  $$\sin \theta = \frac{y}{r} = \frac{4\sqrt{3}}{8} = \frac{\sqrt{3}}{2}$$
  Since $\cos \theta \lt 0$ and $\sin \theta \gt 0$, the point lies in the Second Quadrant:
  $$\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$$
• Polar form:
  $$z = 8 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$$
• Incorrect: $\pi/3$ lies in the first quadrant ($4 + i4\sqrt{3}$).
• Incorrect: $\pi/6$ corresponds to argument of $4\sqrt{3} + i4$.
• Correct: Accurately gives $r = 8$ and $\theta = 2\pi/3$.
• Incorrect: $5\pi/6$ corresponds to $-4\sqrt{3} + i4$.

Analysis of set theory laws:

• Incorrect: $(A \cap B)^C = A^C \cup B^C$, not $(A \cup B)^C$.
• Correct: By De Morgan's first law of complementation, the complement of the union of two sets is the intersection of their complements: $(A \cup B)^C = A^C \cap B^C$.
• Incorrect: $A^C \cup B^C = (A \cap B)^C$.
• Incorrect: $A \cup B$ is the set itself, not its complement.