Let X = (a, b, c, d) and TX = (Ø, X, (b), {b.c), {a,b,d}) be a topology 3, 4) and TY = (0, Y, (3), {1,2,3} } be a topology on Y. Let f be a mapping from X into Y defined by f(a)=f(b) 3 3, f(c) = 2 and f(d) = 1. Letg be mapping from X into Y defined by g(a) = 1. g(b)=g(c)=3 and g(d) = 2. Then O fis not continuous andg is continuous O fand g are not continuotis O fand g are both continuous O fis continuous and g is not continuous

Let X = (a, b, c, d) and TX = (Ø, X, (b), {b.c), {a,b,d}) be a topology 3, 4) and TY = (0, Y, (3), {1,2,3} } be a topology on Y. Let f be a mapping from X into Y defined by f(a)=f(b) 3 3, f(c) = 2 and f(d) = 1. Letg be mapping from X into Y defined by g(a) = 1. g(b)=g(c)=3 and g(d) = 2. Then O fis not continuous andg is continuous O fand g are not continuotis O fand g are both continuous O fis continuous and g is not continuous

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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abstract
Task 1 Verify that a given mapping L is linear if: L(x,y, z)=(3*,y, 2z),x, y, z E R. If L is a mapping then determine a) Im L b) Ker L
Let X = {a, b, c, d} and TX = {Ø, X, {b}, {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3, 4} and TY = {Ø, Y, {3}, {1,2,3}} %3D be a topology on Y. Let f be a mapping from X into Y defined by f(a)=f(b) = 3, f(c) = 2 and f(d) = 1. Let g be mapping from X into Y defined by g(a) = 1, g(b)=g(c)=3 and g(d) = 2. Then * f is continuous and g is not continuous f is not continuous and g is continuous f and g are not continuous f and g are both continuous
Problem 2 Let T be the relation on R x R defined by (x, y) T (a, b) if x² - y = a²-b. Part (a): Prove that T is an equivalence relation on RX R. Part (b): Sketch the equivalence class of (2,1) (this will be a curve in the plane).
Show how a projective transformation in â: R°/~ → R°/ induces an affine map a if and only if â maps the line at infinity to itself. Prove that â induces a translation a if and only if â fixes each point on the line at infinity.
Prove that the set V = {(x, y) : x, y e R} of all ordered pairs with the operations (x, y) © (x', y') = (x + a' + 1, y + y' + 1) for (x, y), (a', y') e V and s0 (r, y) (s +sx-1, s + sy – 1) for s e R is a vector space. Then write (4,0) e V as a linear combination of (1,0) and (0, 1).
2. Let A: V→ V be a linear map. Show that the following statements are equivalent: (a) 0 is an isolated fixed point of A. (b) A – I:V –→ Vis an isomorphism. (c) 0 is a Lefschetz fixed point of A. (d) A is a Lefschetz map.
.Let X and Y be Banach spaces and let T/X -> Y be a continuous (bounded) linear transformation from Xb onto Y. Then T(A) is open set in Y for each open set A of X, i.e., T is an open mapping.
Let X = {a, b, c, d} and Y = {x, Y, z, w} be two topological spaces with topologies T = {X,0, {a}, {a, b}, {a, b, c}} T2 = {Y, Ø, {x}, {y}, {x, y}, {y, z, w}}, respectively. Consider the functions f: X → Y and g : X → Y defined by the and diagrams below: 0000 a a b. b d d Determine which function is continuous and which is not. Show your solution.