Do #: 1.13 a, b, c, 1.15 a, b, c, d, 1.19 & 1.22 1.13 For the map f: P1 + R2 given by at barn (9-b) Find the image of each of these elements of the domain. (a) 3-2x (b) 2 + 2x (c) x Show that this map is an isomorphism. 1.14 Show that the natural map f, from Example 1.5 is an isomorphism. ✓ 1.15 Decide whether each map is an isomorphism (if it is an isomorphism...
Let a function f be f: R rightarrow R such that f(x) = 0 when x lessthanorequalto 0 and f(x) = log(x) * sin(x) when x > 0. f is plotted in the figure below. a) Determine whether f is one-to-one. b) Determine whether f is onto. c) Determine whether f is total. d) Determine ranges for x and y so that f is total but not one-to-one. e) Determine ranges for x and y so that f is one-to-one,...
5. Let A =R x R and f: A+ A be given by the rule f(x, y) = (x – y, x + y). (a) Prove f is one-to-one. (b) Prove f is onto A. (Comment: don't forget that if given b E A, you construct a such that f(a) = b, you must also show a E A.) (c) What is the inverse function? (d) Is f a permutation? Explain.
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
Please do 2 only please do 2 only Exercises (1) Compute for de and c ) da where is the ultime center at the origin and oriented once in the counterclockwise (2) Computer da, where I is the circle {: € C: 1:= 3) once in the counterclockwise direction (3) (Mean Value Property of Holomorphic Functions) Supposed w = f(e) is holomorphic on and inside the circle {: € C:- Prove that f(20) == f( 70 +re) de. (4) Under...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
11.) Let T:R" - R"be a linear transformation. Prove T is onto if and only if T is one-to-one. 12.) Let T:R" - R" and S:R" - R" be linear transformations such that TSX=X for all x ER". Find an example such that ST(x))+x for some xER". - .-.n that tidul,
PLEASE do parts a, b, and c. Thank you. Let f: R S be a homomorphism of rings. Let J be an ideal in S. Let I = {r E R : f(r) € J}. la Prove: ker f +$. « Prove: ker f SI. Prove: I is an ideal in R.