If A and B are sets and f: A→ B, then for any subset S of A we define f(S) = {be B: b= f(a) for some a € S}. Similarly, for any subset T of B we define the pre-image of T as f(T) = {ae A: f(a) € T}. Note that f-¹(T) is well defined even if f does not have an inverse! For each of the following state whether it is True or False. If True then give a proof. If False then give a counterexample: (a) f(S₁US₂) = f(S₁) u f(S₂) (b) f(Sin S₂) = f(S₁) nf (S₂) (c) f¹(T₁UT₂) = f¹(T₁)uf-¹(T₂) (d) f-¹(T₁T₂) = f-¹(T₁) nf-¹(T₂)
If A and B are sets and f: A→ B, then for any subset S of A we define f(S) = {be B: b= f(a) for some a € S}. Similarly, for any subset T of B we define the pre-image of T as f(T) = {ae A: f(a) € T}. Note that f-¹(T) is well defined even if f does not have an inverse! For each of the following state whether it is True or False. If True then give a proof. If False then give a counterexample: (a) f(S₁US₂) = f(S₁) u f(S₂) (b) f(Sin S₂) = f(S₁) nf (S₂) (c) f¹(T₁UT₂) = f¹(T₁)uf-¹(T₂) (d) f-¹(T₁T₂) = f-¹(T₁) nf-¹(T₂)
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
Related questions
Question
2.8
![If A and B are sets and f: A→ B, then for any subset S of A we define
f(S) = {be B: b= f(a) for some a € S}.
Similarly, for any subset T of B we define the pre-image of T as
f(T) = {ae A: f(a) e T}.
Note that f-¹(T) is well defined even if f does not have an inverse!
For each of the following state whether it is True or False. If True then give a proof. If False
then give a counterexample:
(a) f(S₁US₂) = f(S₁) u f(S₂)
(b) f(Sin S₂) = f(S₁) nf (S₂)
(c) f¹(T₁UT₂) = f¹(T₁)uf-¹(T₂) (d) f-¹(T₁T₂) = f-¹(T₁) nf-¹(T₂)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F971b8674-8a06-4fc2-8fd8-5cb998cbb624%2F6636d89c-8e00-43f2-aac8-7091704b79a6%2Fzltwuz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:If A and B are sets and f: A→ B, then for any subset S of A we define
f(S) = {be B: b= f(a) for some a € S}.
Similarly, for any subset T of B we define the pre-image of T as
f(T) = {ae A: f(a) e T}.
Note that f-¹(T) is well defined even if f does not have an inverse!
For each of the following state whether it is True or False. If True then give a proof. If False
then give a counterexample:
(a) f(S₁US₂) = f(S₁) u f(S₂)
(b) f(Sin S₂) = f(S₁) nf (S₂)
(c) f¹(T₁UT₂) = f¹(T₁)uf-¹(T₂) (d) f-¹(T₁T₂) = f-¹(T₁) nf-¹(T₂)
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Recommended textbooks for you
![Database System Concepts](https://www.bartleby.com/isbn_cover_images/9780078022159/9780078022159_smallCoverImage.jpg)
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
![Starting Out with Python (4th Edition)](https://www.bartleby.com/isbn_cover_images/9780134444321/9780134444321_smallCoverImage.gif)
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
![Digital Fundamentals (11th Edition)](https://www.bartleby.com/isbn_cover_images/9780132737968/9780132737968_smallCoverImage.gif)
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
![Database System Concepts](https://www.bartleby.com/isbn_cover_images/9780078022159/9780078022159_smallCoverImage.jpg)
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
![Starting Out with Python (4th Edition)](https://www.bartleby.com/isbn_cover_images/9780134444321/9780134444321_smallCoverImage.gif)
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
![Digital Fundamentals (11th Edition)](https://www.bartleby.com/isbn_cover_images/9780132737968/9780132737968_smallCoverImage.gif)
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
![C How to Program (8th Edition)](https://www.bartleby.com/isbn_cover_images/9780133976892/9780133976892_smallCoverImage.gif)
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
![Database Systems: Design, Implementation, & Manag…](https://www.bartleby.com/isbn_cover_images/9781337627900/9781337627900_smallCoverImage.gif)
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
![Programmable Logic Controllers](https://www.bartleby.com/isbn_cover_images/9780073373843/9780073373843_smallCoverImage.gif)
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education