Let f: X → Y and g : Y → Z be functions. (a) Show that if g of is one-to-one (or "injective"), then f is one-to-one. (b) Give an example of spaces X, Y, Z, and functions f: X → Y and g: Y Z, such that that g of is one-to-one, but g is not one-to-one. Let X = Y = Z = Define f and g as follows: (c) Suppose g of is one-to-one and f is onto (surjective). Show that g is one-to-one.
Let f: X → Y and g : Y → Z be functions. (a) Show that if g of is one-to-one (or "injective"), then f is one-to-one. (b) Give an example of spaces X, Y, Z, and functions f: X → Y and g: Y Z, such that that g of is one-to-one, but g is not one-to-one. Let X = Y = Z = Define f and g as follows: (c) Suppose g of is one-to-one and f is onto (surjective). Show that g is one-to-one.
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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Transcribed Image Text:Let f: X → Y and g : Y → Z be functions.
(a) Show that if g o f is one-to-one (or "injective"), then f is one-to-one.
(b) Give an example of spaces X, Y, Z, and functions f: X → Y and g: Y → Z, such
that that g of is one-to-one, but g is not one-to-one.
Let X =
Y =
Z =
Define f and g as follows:
(c) Suppose g of is one-to-one and f is onto (surjective). Show that g is one-to-one.
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