Let f : A rightarrow D and g : B rightarrow C be functions. For each part, if the answer is yes, then prove it, otherwise give a counterexample. Suppose f is one-to-one (injective) and g is onto (surjective). Is go f one-to-one (injective)? Suppose f is one-to-one (injective) and g is onto (surjective). Is g f onto (surjective)? Suppose g is one-to one. Is g one-to-one? Suppose g f onto. Is g onto?
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is...
Problem 1.3. For each function fi, determine whether it is injective but not surjective, surjective but not injective, bijective, or neither injective nor surjective. Explain why. (1) f1: R20 + R with f1(x) = x2 for all x ER>, where R20 = {x ER|X>0} = [0, ). (2) f2: R20 + R20 with f2(x) = x2 for all c ER>0. (3) f3: R + Ryo with f3(2) = x4 for all x € R. (4) f4: R R with f4(:1)...
5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that Gal( E/O) Q and that Fi and F2 generate E is a field isomorphic to the product Gal(F/Q) x Gal(F2/Q) is X 5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that Gal( E/O) Q and that Fi and F2...
PART B ONLY. Let fi be a continuous function with different signs at a, b, with a < b and let nn be bisection method's sequence of approximations on f using starting interval [a, b]. Let f2 be a continuous function with different signs at a, b, with a 〈 b and let {dn} 0 be bisection method's sequence of approximations on f2 using starting interval [a, b (a) Prove (perhaps by induction) if ckd, for some k, then cd,...
Let fi be a continuous function with different signs at a, b, with a < band let (cn be bisection method's sequence of approximations on f using starting interval a, b. Let f2 be a continuous function with different signs at a, b, with a< b and let dnn be bisection method's sequence of approximations on f2 using starting interval a, b (a) Prove (perhaps by induction) if cdk, for some k, then c d, for all i < k....
For nonempty sets A, B and C, let f : A → B and g : B → C be functions. Prove that if g ◦ f is injective, then f is injective
Let fi and f2 be functions such that lim e s f1 (2) = + and such that the limit L2 = lim a s f2 (x) exists. Which one of the following is NOT correct? O limas (f1f2)(x) = 0 if L2 = 0. limas (fi + f2)(x) = too if L2 = -0. Olim as (f1f2) (x) = too if 0 <L2 5+co. lim a s (f1f2)(x) = - it L2 = -. Which one of the following...
Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
Please help me to prove this proposition! Thanks a lot! Proposition Let fi, f2 be power series centered, respectively, at 21, 22 with radius of convergence R1, R2. Suppose that Dri (21)NDR2 (22) # 0 and that fi, f2 agree on the overlap DR1 (zı)nDR2(22). Then, fı = f2.