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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
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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.
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