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Exercise 3.12. Let A B be a one-to-one correspondence. 1. Prove that l is a function. 2. Prove that f- is one-to-one. 3. Prove that - is onto. 4. Conclude that fl: B A is a one-to-one correspondence.

Exercise 3.12. Let A B be a one-to-one correspondence. 1. Prove that l is a function. 2. Prove that f- is one-to-one. 3. Prove that - is onto. 4. Conclude that fl: B A is a one-to-one correspondence.

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 help me with Exercice 3.12
Molepo/Desktop/UNISA/MAT2611%20(Linear%20Algebra%2011)/101_2022_0_b(2).pdf G Page view A Read aloud A Add text - Draw F Highlic 1.:RR defined by f(2) = 4x - 15. 2. g: R+Rdefined by f(r) = 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 be B there corresponds a unique a E A such that f(a) = b. We define f-1: B A by 6) = the unique a such that f(a) = b. Exercise 3.12. Let fA B be a one-to-one correspondence. 1. Prove that f-1 is a function. 2. Prove that f-1 is one-to-one. 3. Prove that -1 is onto. 4. Conchide that f1: B→ A is a one-to-ond correspondence. Exercise 3.13. Let f: A B be a one-to-one correspondence. By Exercise 3.12, B A is also a one-to-one correspondence. 1. Prove that fof =i4. 2. Prove that o = iB. Set equivalence 3.6 We are finally in a position to give a formnal definition of the size of a set and to compare different sizes of sets. Informally speaking, if f : A B is a one-to-one function, then since different elements of A are mapped to different clements of B, the size of B is at lcast as large as the size of A. On the other hand if f is onto then since each element in B has at least one element in A that is mapped to it, the size of B is no greater than the size of A. Thus, one-to-one correspondences provide as with a means to compare the sizes of sets. This key observation of Cantor led him to the notion of two sets being equivalear. Let us read how Cantor defines that two sets are equivalent. We say that two aggregates V and Nare requiva lent." in signs
Transcribed Image Text:Molepo/Desktop/UNISA/MAT2611%20(Linear%20Algebra%2011)/101_2022_0_b(2).pdf G Page view A Read aloud A Add text - Draw F Highlic 1.:RR defined by f(2) = 4x - 15. 2. g: R+Rdefined by f(r) = 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 be B there corresponds a unique a E A such that f(a) = b. We define f-1: B A by 6) = the unique a such that f(a) = b. Exercise 3.12. Let fA B be a one-to-one correspondence. 1. Prove that f-1 is a function. 2. Prove that f-1 is one-to-one. 3. Prove that -1 is onto. 4. Conchide that f1: B→ A is a one-to-ond correspondence. Exercise 3.13. Let f: A B be a one-to-one correspondence. By Exercise 3.12, B A is also a one-to-one correspondence. 1. Prove that fof =i4. 2. Prove that o = iB. Set equivalence 3.6 We are finally in a position to give a formnal definition of the size of a set and to compare different sizes of sets. Informally speaking, if f : A B is a one-to-one function, then since different elements of A are mapped to different clements of B, the size of B is at lcast as large as the size of A. On the other hand if f is onto then since each element in B has at least one element in A that is mapped to it, the size of B is no greater than the size of A. Thus, one-to-one correspondences provide as with a means to compare the sizes of sets. This key observation of Cantor led him to the notion of two sets being equivalear. Let us read how Cantor defines that two sets are equivalent. We say that two aggregates V and Nare requiva lent." in signs
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