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a) Let O be the set of odd numbers and O' = {1, 5, 9, 13, 17, ...} be its subset. Define the bijections, f and g as: f: 0-0', f(d) = 2d - 1, Vd € 0. g: N- O, g(n) = 2n + 1, Vn € N. Using only the concept of function composition, can there be a bijective map from N to O'? If so, compute it. If not, explain in details why not.

a) Let O be the set of odd numbers and O' = {1, 5, 9, 13, 17, ...} be its subset. Define the bijections, f and g as: f: 0-0', f(d) = 2d - 1, Vd € 0. g: N- O, g(n) = 2n + 1, Vn € N. Using only the concept of function composition, can there be a bijective map from N to O'? If so, compute it. If not, explain in details why not.

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
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a) Let O be the set of odd numbers and O' = {1, 5, 9, 13, 17, ..} be its subset. Define the bijections, f and g as: f:0 - 0', f(d) = 2d - 1, Vd e O. g:N - 0, g(n) = 2n + 1, Vn e N. Using only the concept of function composition, can there be a bijective map from N to O'? If so, compute it. If not, explain in details why not.
Transcribed Image Text:a) Let O be the set of odd numbers and O' = {1, 5, 9, 13, 17, ..} be its subset. Define the bijections, f and g as: f:0 - 0', f(d) = 2d - 1, Vd e O. g:N - 0, g(n) = 2n + 1, Vn e N. Using only the concept of function composition, can there be a bijective map from N to O'? If so, compute it. If not, explain in details why not.
b) A Sesotho word cannot begin with of the following letters of alphabet: D, G, V, W, X, Y and Z. We define the relation: A Sesotho word x is related to another Sesotho word y if x begins with the same letter as y. Determine whether or not this is an equivalence relation. If it is an equivalence relation then 1. Compute C(sekatana) 2. How many equivalence classes are there in all, and why? 3. What is the partition of the English words under this relation? If it is NOT an equivalence relation then explain in details why it is not.
Transcribed Image Text:b) A Sesotho word cannot begin with of the following letters of alphabet: D, G, V, W, X, Y and Z. We define the relation: A Sesotho word x is related to another Sesotho word y if x begins with the same letter as y. Determine whether or not this is an equivalence relation. If it is an equivalence relation then 1. Compute C(sekatana) 2. How many equivalence classes are there in all, and why? 3. What is the partition of the English words under this relation? If it is NOT an equivalence relation then explain in details why it is not.
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