Hey. Please help me with Exercise 3.13 LM.Molepo/Desktop/UNISA/MAT2611%20(Linear%20Algebra%201)/101_2022_0_b(2).pdfO L Page viewA Read aloudE Add textV DrawE Highl1. f:R+ R defined by f(x) = 4r - 15.%3D2. 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 uniquea E A such that f(a) = b. We define f-1: B A byf(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 isalso a one-to-one correspondence.1. Prove that f of =i2. Prove that fof-1 = iB.3.6Set equivalenceWe are finally in a position to give a formal definition of the size of a set and to compare differentsizes of sets. Informally speaking, if felements of A are mapped to different elements of B, the size of B is at least as large as the size ofA. On the other hand, if f is onto, then since each element in B has at least one element in A thatis mapped to it, the size of B is no greater than the size of A. Thus, one-to-one correspondencesprovide us with a means to compare the sizes of sets. This key ohservation of Cantor led him to thenotion 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 differentWe say that two aggregates M and N are "equivalent," in signsNor N N

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Hey. Please help me with Exercise 3.13 LM.Molepo/Desktop/UNISA/MAT2611%20(Linear%20Algebra%201)/101_2022_0_b(2).pdfO L Page viewA Read aloudE Add textV DrawE Highl1. f:R+ R defined by f(x) = 4r - 15.%3D2. 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 uniquea E A such that f(a) = b. We define f-1: B A byf(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 isalso a one-to-one correspondence.1. Prove that f of =i2. Prove that fof-1 = iB.3.6Set equivalenceWe are finally in a position to give a formal definition of the size of a set and to compare differentsizes of sets. Informally speaking, if felements of A are mapped to different elements of B, the size of B is at least as large as the size ofA. On the other hand, if f is onto, then since each element in B has at least one element in A thatis mapped to it, the size of B is no greater than the size of A. Thus, one-to-one correspondencesprovide us with a means to compare the sizes of sets. This key ohservation of Cantor led him to thenotion 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 differentWe say that two aggregates M and N are &#34;equivalent,&#34; in signsNor N N

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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.