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9. Recall that Q denotes the set of all rational numbers and Z denotes the set of all integers. Consider the functions h : Z → Z given by h(x) 3x + 5 and h : Q → Q given by h(x) = 3x + 5. Show that h is not surjective but h is surjective although both functions are defined by the expression 3x + 5.

9. Recall that Q denotes the set of all rational numbers and Z denotes the set of all integers. Consider the functions h : Z → Z given by h(x) 3x + 5 and h : Q → Q given by h(x) = 3x + 5. Show that h is not surjective but h is surjective although both functions are defined by the expression 3x + 5.

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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9. Recall that \( \mathbb{Q} \) denotes the set of all rational numbers and \( \mathbb{Z} \) denotes the set of all integers. Consider the functions \( h : \mathbb{Z} \rightarrow \mathbb{Z} \) given by \( h(x) = 3x + 5 \) and \( \tilde{h} : \mathbb{Q} \rightarrow \mathbb{Q} \) given by \( \tilde{h}(x) = 3x + 5 \). Show that \( h \) is not surjective but \( \tilde{h} \) is surjective although both functions are defined by the expression \( 3x + 5 \).
Transcribed Image Text:9. Recall that \( \mathbb{Q} \) denotes the set of all rational numbers and \( \mathbb{Z} \) denotes the set of all integers. Consider the functions \( h : \mathbb{Z} \rightarrow \mathbb{Z} \) given by \( h(x) = 3x + 5 \) and \( \tilde{h} : \mathbb{Q} \rightarrow \mathbb{Q} \) given by \( \tilde{h}(x) = 3x + 5 \). Show that \( h \) is not surjective but \( \tilde{h} \) is surjective although both functions are defined by the expression \( 3x + 5 \).
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