Homework Help is Here – Start Your Trial Now!
Engineering
Computer Science
7. Let X be a set of cardinality n > 0. Let E be the set of all subsets of X with even cardinality and O the set of all subsets of X with odd cardinality. Let x be any element of the set X and let ƒ be the mapping from E to O given by the following rule: for all A Є E, let ƒ(A) = A \ {x} if x is an element of A, and f(A) = AU{x} otherwise. (i) Prove that f is bijective. (ii) Using the result in (i), give a combinatorial proof of the equality Σ 3-0 (−1)³ (n) = 0.

7. Let X be a set of cardinality n > 0. Let E be the set of all subsets of X with even cardinality and O the set of all subsets of X with odd cardinality. Let x be any element of the set X and let ƒ be the mapping from E to O given by the following rule: for all A Є E, let ƒ(A) = A \ {x} if x is an element of A, and f(A) = AU{x} otherwise. (i) Prove that f is bijective. (ii) Using the result in (i), give a combinatorial proof of the equality Σ 3-0 (−1)³ (n) = 0.

Database System Concepts
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
See similar textbooks
icon
Related questions
Question
7. Let X be a set of cardinality n > 0. Let E be the set of all subsets of X with even cardinality and O the set of all subsets of X with odd cardinality. Let x be any element of the set X and let ƒ be the mapping from E to O given by the following rule: for all A Є E, let ƒ(A) = A \ {x} if x is an element of A, and f(A) = AU{x} otherwise. (i) Prove that f is bijective. (ii) Using the result in (i), give a combinatorial proof of the equality Σ 3-0 (−1)³ (n) = 0.
Transcribed Image Text:7. Let X be a set of cardinality n > 0. Let E be the set of all subsets of X with even cardinality and O the set of all subsets of X with odd cardinality. Let x be any element of the set X and let ƒ be the mapping from E to O given by the following rule: for all A Є E, let ƒ(A) = A \ {x} if x is an element of A, and f(A) = AU{x} otherwise. (i) Prove that f is bijective. (ii) Using the result in (i), give a combinatorial proof of the equality Σ 3-0 (−1)³ (n) = 0.
Expert Solution
steps

Step by step

Solved in 1 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Database System Concepts
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)
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS