b) We define a function f: N→ Z such that:피/2if x is evenf(x) =-필 if z is oddi) Prove that f is an injection.i) Prove that f is a bijection.i) Given that f is a bijection, find the inverse function fl: Z →Niv) Given the previous parts, what can we say about the cardinality of N and Z?

Since you have posted multiple subparts we solved first three subparts for you. To get the solution…

b) We define a function f: N→ Z such that: x/2 if x is even -1 if ar is odd f(x) = i) Prove that f is an injection. i) Prove that f is a bijection. ii) Given that f is a bijection, find the inverse function f1: Z→N iv) Given the previous parts, what can we say about the cardinality of N and Z?

b) We define a function f: N→ Z such that: x/2 if x is even -1 if ar is odd f(x) = i) Prove that f is an injection. i) Prove that f is a bijection. ii) Given that f is a bijection, find the inverse function f1: Z→N iv) Given the previous parts, what can we say about the cardinality of N and Z?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 11E

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Question

b) We define a function f: N→ Z such that:
피/2
if x is even
f(x) =
-필 if z is odd
i) Prove that f is an injection.
i) Prove that f is a bijection.
i) Given that f is a bijection, find the inverse function fl: Z →N
iv) Given the previous parts, what can we say about the cardinality of N and Z?

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