Let S and T be non-empty c+s. Considev the pro i a surection ) Show tnat...
4. Let A be a non-empty set and f: A- A be a function. (a) Prove that f has a left inverse in FA if and only if f is injective (one-to-one) (b) Prove that, if f is injective but not surjective (which means that the set A is infinite), then f has at least two different left inverses.
Prove: Let A be a dense subset of (X, T), and let B be a non-empty open subset of X. Then An B Prove: Let A be a dense subset of (X, T), and let B be a non-empty open subset of X. Then An B
(6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S. (6) Let S c R be non-empty and bounded above. Let q = sup S. Show that q E bd S.
Let S be the surface S ((x, y, z) ER3z 7y2 0 (i) Show that the function a :R2-R3, , given by a(t, u)- (t 2,3ut, 7u2), is C1 on all of R2 and satisfies a(t,u) E S for all (t,u) E R2 ii) Show that a is not injective. (ii) Find all the points of the domain where Da is not injective.
1. Let A, B be two non-empty sets and f: A + B a function. We say that f satisfies the o-property if VC+0.Vg, h: C + A, fog=foh=g=h. Prove that f is injective if and only if f satisfies the o-property.
5. Let S be a non empty bounded subset of R. If a > 0, show that sup (as) = a sup S where as = {as : ES}. Let c = sup S, show ac = sup (aS). This is done by showing (a) ac is an upper bound of aS. (b) If y is another upper bound of as then ac S7 Both are done using definitions and the fact that c=sup S.
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
Please show all work so I can gain a better understanding. Thank you! (Let X ⊂ R n be non-empty and let A be an n×n matrix. Show that A[co (X)] = co (A[X]). Here co means convex hull.) Exercise 17: Let X C Rn be non-empty and let A be an n × n matrix. Show that Alco (X)-co (A Here co means convex hull. ) Exercise 17: Let X C Rn be non-empty and let A be an...
5. Let S be a non-empty bounded subset of R. If a > 0, show that sup (aS) = a sup S where aS = {as : s E S}. Let c = sup S, show ac = sup (aS). This is done by showing: (a) ac is an upper bound of aS. (b) If y is another upper bound of aS then ac < 7. Both are done using definitions and the fact that c=sup S.
I. Functions and Isomorphisms. Let G be a group and let a EG be any non-identity element (so a #e). Define a function f : GG so that, for any r EG, f(x) = (xa)-1 (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer. (c) Is f an isomorphism? Prove your answer.