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Exercise 3.13. Letf:A B be a one-to-one correspohde also a one-to-one correspondence. 1. Prove that f-lof=i4. 2. Prove that fof =iB-

Exercise 3.13. Letf:A B be a one-to-one correspohde also a one-to-one correspondence. 1. Prove that f-lof=i4. 2. Prove that fof =iB-

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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Hey. Please help me with Exercise 3.13
p/Desktop/UNISA/MAT2611%20(Linear%20Algebra%2011)/101_2022_0_b(2),pdf IL Page view A Read aloud O Add text V Draw 1. f:R→R defined by f(r) = 4r - 15. 2. g: R R defined by f(z) = 152. Prove that both f and g are one-to-one correspondences. Let f: A B be a one-to-one correspondence. Then to each bEB there corresponds a unique a E A such that f(a) = b. We define f: B-A by (b) = the unique a such that f(a) = b. Exercise 3.12. Let f: A B be a one-to-one corespondence. 1. Prove that f- is a function. 2. Prove that f-1 is one-to-one. 3. Prove that f-1 is onto, 4. Conclude that f: B A is a one-to-one coTTEspondence. Exercise 3.13. Let f: A B be a one-to-oe correspondence. By Exercise 3.12 B- A is also a one-to-one correspondence. 1. Prove that fof =i4. 2. Prove that fof=ig- 3.6 Set equivalence We are finally in a position to give a formal definition of the size of set and to edormpare different sizes of sets. Informally speaking i elements of A are napped to different klements A. On the other hand, iff is onto, then since ee is mapped to it. the size of Bis co grents than t ze of provide ns with a mcans to compare the sizs osets. This key observation of Cantor led hin to the notion of two sets being cquivalerit Leet s read ovW Canter klefines that two sets are cquisalent. function, then since different sat lestas large as the siz of at least oneclement in A that Lheo-one eorrespondences We say that two aggregates Mand Nare equivalent in signs
Transcribed Image Text:p/Desktop/UNISA/MAT2611%20(Linear%20Algebra%2011)/101_2022_0_b(2),pdf IL Page view A Read aloud O Add text V Draw 1. f:R→R defined by f(r) = 4r - 15. 2. g: R R defined by f(z) = 152. Prove that both f and g are one-to-one correspondences. Let f: A B be a one-to-one correspondence. Then to each bEB there corresponds a unique a E A such that f(a) = b. We define f: B-A by (b) = the unique a such that f(a) = b. Exercise 3.12. Let f: A B be a one-to-one corespondence. 1. Prove that f- is a function. 2. Prove that f-1 is one-to-one. 3. Prove that f-1 is onto, 4. Conclude that f: B A is a one-to-one coTTEspondence. Exercise 3.13. Let f: A B be a one-to-oe correspondence. By Exercise 3.12 B- A is also a one-to-one correspondence. 1. Prove that fof =i4. 2. Prove that fof=ig- 3.6 Set equivalence We are finally in a position to give a formal definition of the size of set and to edormpare different sizes of sets. Informally speaking i elements of A are napped to different klements A. On the other hand, iff is onto, then since ee is mapped to it. the size of Bis co grents than t ze of provide ns with a mcans to compare the sizs osets. This key observation of Cantor led hin to the notion of two sets being cquivalerit Leet s read ovW Canter klefines that two sets are cquisalent. function, then since different sat lestas large as the siz of at least oneclement in A that Lheo-one eorrespondences We say that two aggregates Mand Nare equivalent in signs
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