Final answer:
The statement ‘“|z-w| > |z| - |w|’“ is false, as it contradicts the triangle inequality in complex numbers. The correct inequality should be |z-w| ≤ |z| + |w|.
Step-by-step explanation:
The question asks us to prove whether the statement ‘“|z-w| > |z| - |w|’“ given two complex numbers z and w is true or false. To address this, we can use the triangle inequality, which states that for any complex numbers z and w, the inequality |z + w| ≤ |z| + |w| holds true. Here, we can replace w with -w to get: |z + (-w)| ≤ |z| + |-w|, which simplifies to |z - w| ≤ |z| + |w|, since the absolute value of a real number is always non-negative.
The correct answer, therefore, is not that ‘“|z-w| > |z| - |w|’“, but rather that |z-w| ≤ |z| + |w|. It is also important to note that subtraction does not naturally apply to the right side of the inequality as in the original statement. Hence, part of the original question is incorrect, and none of the given options (a), (b), or (c) are true.