Need help with ONLY homework problem 2 in Foundations of Mathematics. I received feedback from my professor for problem 2 and he said "Clearly, = 0 or = 0" No, this is not clear. You need to prove this. Below the homework problems is my work for problem 2. 1. Let \( M \) be a nonempty set and \( d: M^2 \rightarrow \mathbb{R} \) be defined as \( d(x,y) = \begin{cases} 1 & \text{if } x \neq y \\ 0 & \text{if } x = y \end{cases} \). Prove that \( (M,d) \) is a metric space.2. Let \( z_1, z_2 \in \mathbb{C} \). Prove that \( z_1 z_2 = 0 \) if and only if \( z_1 = 0 \) or \( z_2 = 0 \). 2.**Proof**Let \( z_1, z_2 \in \mathbb{C} \).\( z_1 z_2 = 0 \) if and only if \( z_1 = 0 \) or \( z_2 = 0 \).1. Assume \( z_1 = 0 \) or \( z_2 = 0 \).2. Assume \( z_1 z_2 \neq 0 \); \( z_1, z_2 \in \mathbb{C} \). * (On the contrary) * \( \Rightarrow z_1 \neq 0 \) and \( z_2 \neq 0 \) * which is a contradiction.Thus, \( z_1 z_2 = 0 \).Conversely, let \( z_1 z_2 = 0 \rightarrow (\ast) \).Clearly, \( z_1 = 0 \) or \( z_2 = 0 \).Suppose \( z_1 \neq 0 \Rightarrow z_2 = 0 \) (by \( (\ast) \))or\( z_2 \neq 0 \Rightarrow z_1 = 0 \) (by \( (\ast) \)).

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Answered: 2. Let 21 22 € C. Prove that 21 22 =…

Math

Advanced Math

2. Let 21 22 € C. Prove that 21 22 = 0iff z₁ = 0 or 22 = 0.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Need help with ONLY homework problem 2 in Foundations of Mathematics. I received feedback from my professor for problem 2 and he said "Clearly, = 0 or = 0" No, this is not clear. You need to prove this. Below the homework problems is my work for problem 2.

1. Let \( M \) be a nonempty set and \( d: M^2 \rightarrow \mathbb{R} \) be defined as \( d(x,y) = \begin{cases}
1 & \text{if } x \neq y \\
0 & \text{if } x = y
\end{cases} \). Prove that \( (M,d) \) is a metric space.

2. Let \( z_1, z_2 \in \mathbb{C} \). Prove that \( z_1 z_2 = 0 \) if and only if \( z_1 = 0 \) or \( z_2 = 0 \).

2.

**Proof**

Let \( z_1, z_2 \in \mathbb{C} \).

\( z_1 z_2 = 0 \) if and only if \( z_1 = 0 \) or \( z_2 = 0 \).

1. Assume \( z_1 = 0 \) or \( z_2 = 0 \).

2. Assume \( z_1 z_2 \neq 0 \); \( z_1, z_2 \in \mathbb{C} \).

* (On the contrary)

* \( \Rightarrow z_1 \neq 0 \) and \( z_2 \neq 0 \)

* which is a contradiction.

Thus, \( z_1 z_2 = 0 \).

Conversely, let \( z_1 z_2 = 0 \rightarrow (\ast) \).

Clearly, \( z_1 = 0 \) or \( z_2 = 0 \).

Suppose \( z_1 \neq 0 \Rightarrow z_2 = 0 \) (by \( (\ast) \))

or

\( z_2 \neq 0 \Rightarrow z_1 = 0 \) (by \( (\ast) \)).

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