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Exercise 3.13. Let f: A+B be a one-to-one correspondence. By Exercise 3.12, f-: B A is also a one-to-one correspondence. 1. Prove that fof =in. 2. Prove that fof-l = iB.

Exercise 3.13. Let f: A+B be a one-to-one correspondence. By Exercise 3.12, f-: B A is also a one-to-one correspondence. 1. Prove that fof =in. 2. Prove that fof-l = 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
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LM.Molepo/Desktop/UNISA/MAT2611%20(Linear%20Algebra%201)/101_2022_0_b(2).pdf O L Page view A Read aloud E Add text V Draw E Highl 1. f:R+ R defined by f(x) = 4r - 15. %3D 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 bE B there corresponds a unique a E A such that f(a) = b. We define f-1: B A by f(b) = the unique a such that f(a) = b. Exercise 3.12. Let f: A + 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 f-1 is onto. 4. Conclude that f-1: B+ A is a one-to-one correspondence. Exercise 3.13. Let f: A + B be a one-to-one correspondence. By Exercise 3.12, f-: B A is also a one-to-one correspondence. 1. Prove that f of =i 2. Prove that fof-1 = iB. 3.6 Set equivalence We are finally in a position to give a formal definition of the size of a set and to compare different sizes of sets. Informally speaking, if f elements of A are mapped to different elements of B, the size of B is at least 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 us with a means to compare the sizes of sets. This key ohservation of Cantor led him to the notion of two sets being equivalent. Let us read how Cantor defines that two sets are equivalent. : A B is a one-to-one function, then since different We say that two aggregates M and N are "equivalent," in signs Nor N N
Transcribed Image Text:LM.Molepo/Desktop/UNISA/MAT2611%20(Linear%20Algebra%201)/101_2022_0_b(2).pdf O L Page view A Read aloud E Add text V Draw E Highl 1. f:R+ R defined by f(x) = 4r - 15. %3D 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 bE B there corresponds a unique a E A such that f(a) = b. We define f-1: B A by f(b) = the unique a such that f(a) = b. Exercise 3.12. Let f: A + 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 f-1 is onto. 4. Conclude that f-1: B+ A is a one-to-one correspondence. Exercise 3.13. Let f: A + B be a one-to-one correspondence. By Exercise 3.12, f-: B A is also a one-to-one correspondence. 1. Prove that f of =i 2. Prove that fof-1 = iB. 3.6 Set equivalence We are finally in a position to give a formal definition of the size of a set and to compare different sizes of sets. Informally speaking, if f elements of A are mapped to different elements of B, the size of B is at least 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 us with a means to compare the sizes of sets. This key ohservation of Cantor led him to the notion of two sets being equivalent. Let us read how Cantor defines that two sets are equivalent. : A B is a one-to-one function, then since different We say that two aggregates M and N are "equivalent," in signs Nor N N
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