4. Working with functions. In this question, we will explore various properties of functions. You may want to review the basic definitions and terminology introduced on pages 15-16 of the course notes. Then, read the following definitions carefully. Definition: A function f : A → B is one-to-one iff no two elements of A have the same image. Symbol- ically, Va1, az E A, f(a1) = f(a2) → a1 = a2. (3) Definition: A function f : A → B is onto iff every element of B is the image of at least one element from A. Symbolically, Vb E B, 3a € A, f (a) = b. (4) Definition: For all functions f : A → B and g : B → C, their composition is the function gof : A → C defined by: Va e A, (g o f)(a) = g(f(a)). (5) Let f : A → B and g : B → C be arbitrary functions. Prove or disprove each of the (c) following. In each case, first write down in symbolic notation the exact statement you are attempting to prove (either the original statement or its negation). i. If go f is one-to-one, then f is also one-to-one. ii. If go f is onto, then g is also onto. iii. If go f is both one-to-one and onto, then f and g are also both one-to-one and onto.
4. Working with functions. In this question, we will explore various properties of functions. You may want to review the basic definitions and terminology introduced on pages 15-16 of the course notes. Then, read the following definitions carefully. Definition: A function f : A → B is one-to-one iff no two elements of A have the same image. Symbol- ically, Va1, az E A, f(a1) = f(a2) → a1 = a2. (3) Definition: A function f : A → B is onto iff every element of B is the image of at least one element from A. Symbolically, Vb E B, 3a € A, f (a) = b. (4) Definition: For all functions f : A → B and g : B → C, their composition is the function gof : A → C defined by: Va e A, (g o f)(a) = g(f(a)). (5) Let f : A → B and g : B → C be arbitrary functions. Prove or disprove each of the (c) following. In each case, first write down in symbolic notation the exact statement you are attempting to prove (either the original statement or its negation). i. If go f is one-to-one, then f is also one-to-one. ii. If go f is onto, then g is also onto. iii. If go f is both one-to-one and onto, then f and g are also both one-to-one and onto.
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:4.
Working with functions. In this question, we will explore various properties of functions.
You may want to review the basic definitions and terminology introduced on pages 15-16 of the course
notes. Then, read the following definitions carefully.
Definition: A function f : A → B is one-to-one iff no two elements of A have the same image. Symbol-
ically,
Va1, a2 E A, f(a1) = f(a2) = a1 = a2.
(3)
%3D
Definition: A function f : A → B is onto iff every element of B is the image of at least one element
from A. Symbolically,
Vb E B, 3a E A, f(a) = b.
(4)
Definition: For all functions f : A → B and g : B → C, their composition is the function g of : A→ C
defined by:
Va E A, (g o f)(a) = g(f(a)).
(5)
Let f : A → B and g : B →→ C be arbitrary functions. Prove or disprove each of the
(c)
following. In each case, first write down in symbolic notation the exact statement you are attempting
to prove (either the original statement or its negation).
i. If gof is one-to-one, then f is also one-to-one.
ii. If gof is onto, then g is also onto.
iii. If go f is both one-to-one and onto, then f and g are also both one-to-one and onto.
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