Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
2. Let 21 22 € C. Prove that 21 22 = 0iff z₁ = 0 or 22 = 0.

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
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 \( z_2 = 0 \).
Transcribed Image Text: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 \( (\ast) \)).
Transcribed Image Text: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 \( (\ast) \)).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 4 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS