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Theorem 13.6 Consider the function f:A – B. Let Uj and Uz be arbitrary subsets of the domain A, and let Vị and V2 be arbitrary subsets of the target space B. Then, (a) if U, C Uz, f(U)C {(U); (b) if V, C V3, ƒ¯'(V,) C f-'(V:); (c) for all sets U, UC F({(U); (d) for all sets V, f(s-'(V) C V; and (e) for all sets V. f-'(V') = (f-"'(V)".

Theorem 13.6 Consider the function f:A – B. Let Uj and Uz be arbitrary subsets of the domain A, and let Vị and V2 be arbitrary subsets of the target space B. Then, (a) if U, C Uz, f(U)C {(U); (b) if V, C V3, ƒ¯'(V,) C f-'(V:); (c) for all sets U, UC F({(U); (d) for all sets V, f(s-'(V) C V; and (e) for all sets V. f-'(V') = (f-"'(V)".

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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Please prove (b) and (e) in typefont and explain each step. For (b), please also show that there cannot be an equality. Thank you!

Theorem 13.6 Consider the function f:A – B. Let Uj and Uz be arbitrary subsets of the domain A, and let Vị and V2 be arbitrary subsets of the target space B. Then, (a) if U, C Uz, f(U)C {(U); (b) if V, C V3, ƒ¯'(V,) C f-'(V:); (c) for all sets U, UC F({(U); (d) for all sets V, f(s-'(V) C V; and (e) for all sets V. f-'(V') = (f-"'(V)".
Transcribed Image Text:Theorem 13.6 Consider the function f:A – B. Let Uj and Uz be arbitrary subsets of the domain A, and let Vị and V2 be arbitrary subsets of the target space B. Then, (a) if U, C Uz, f(U)C {(U); (b) if V, C V3, ƒ¯'(V,) C f-'(V:); (c) for all sets U, UC F({(U); (d) for all sets V, f(s-'(V) C V; and (e) for all sets V. f-'(V') = (f-"'(V)".
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