8. Prove or disprove the following:(a) If X and Y are path-connected, then X x Y is path-connected.(b) If A C X is path-connected, then A is path-connected.(c) If X is locally path-connected, and AC X, then A is locally path-connected.(d) If X is path-connected, and f: X Y is continuous, then f(X) is path-connected.(e) If X is locally path-connected, and f: X→ Y is continuous, then f(X) islocally path-connected.

Let X and Y be topological spaces. We need to prove or disprove the following: If X and Y are…

Answered: Prove or disprove the following: (a) If…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.1: Polynomial Functions Of Degree Greater Than

Problem 54E

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1. Suppose f: A B and W and X are subsets of A. (a) Will it always be true that f(W U X) = f(W) U f(X)? (b) Will it always be true that f(W\ X) = f(W) \ f(X)?
Determine whether the following sets are linearly dependent or linearlyindependent. (a) {(; )( )} . -9} in Maxa(R) 4 (b) {(- ).(- 9G -)} in M2x2(R) 4 (c) {x 3 + 2x2, -x2 + 3x + 1, x3 – x2 + 2x – 1} in P3(R) (d) {x 3 – x, 2x2 + 4, -2x3 + 3x2 + 2x + 6} in P3(R) (e) {(1, -1, 2), (1, -2, 1), (1, 1, 4)} in R3 (f) {(1, –1, 2), (2, 0, 1), (–1, 2, –1)} in R3 {(÷ ) ( -) (; 0. 2 ).(- )} in Max2(R) (g) (m) {(-; ') (? ). -2)} in M2x2(R) () (x 4 - х3 + 5x2 - 8х + 6, -х4 + х3 - 5х2 + 5х - 3, x 4 + 3x2 – 3x + 5, 2x4 + 3x3 + 4x2 – x + 1, x3 – x + 2} in P4(R) ) (x4 - х3 + 5x2 - 8х + 6, -х4 + x3 - 5x2 + 5х - 3, х4 + 3x2 - 3х + 5, 2х4 + х3 + 4x2 + 8х} in P4(R)
3. Find ƒ(2), fƒ(3), ƒ(4), and f(5) if f is defined recur- sively by f(0) = -1, f(1) = 2, and for n = 1, 2, ... a) f(n+1) = f(n) +3f(n-1). b) f(n + 1) = f(n)² ƒ (n − 1). c) f(n+1) = - 3ƒ (n)² – 4ƒ (n − 1)². d) f(n + 1) = f(n − 1)/ƒ (n).
Show that if x is in both W and W-, then x = 0.
2. Determine whether each of these functions from {a, b, c, d} to itself is one-to-one/onto. (a) f(a) = b, f(b) = a, f(c) = c, f(d) = d (b) f(a) = b, f(b) = b, f(c) = d, f(d) = c (c) f(a) = d, f(b) = b, f(c) = c, f(d) = d %3D
4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C T. (b) Under what condition does equality hold?
4. Suppose g : R → R is continuous.Can we say whether the set S = {x : g(x) > 0} is open? closed? neither?
4. Suppose g:R → R is continuous.Can we say whether the set S = {x : g(x) > 0} is open? closed? neither?
1. Consider a function : D→ R where DCR is measurable. Show that the following are equivalent: (i) is measurable. (ii) -¹(O) is measurable for every open set OCR. (iii) ¹(B) is measurable for every Borel set BC R.
1) Let F {Be B(R) | B = -B}.Show that B is a σfeld.
18. Let A = = {W,Y, Z}, C = {r, V2, V5}. Find examples of two functions f: A→ B, and g : B → C, such that {1,2, 3}, B || (a) f is one-to-one, and yet g of is not.
8. Decide which of the following sets are linearly independent in R. Justify your answer in each case. (a) X₁ = {(1,0), (0, 1)} CR² (b) X₂ = {(1,0), (2,0)} CR² (c) X3 = {(-1,0), (0,0)} CR² (d) X₁ = {(1,0), (0, 1), (1, 1)} CR² (e) X5 = {(1,0,0), (0, 1, 0), (1, 1, 1)} CR³ (f) X6 = {(1,0,0), (0, 1, 0), (1, 1, 1), (-1,0,1)} CR³ (g) X7 = {(0, 0, 0), (0, 1, 0), (1, 1, 1), (-1,0,1)} CR³ (h) Xs = {(0, 1, 0), (1,1,1)} CR³ (i) X9 = {(4,3,0,0), (0, 0, 1, 1), (0, 0, 0, 1), (1, 0, 0, 1), (0, 1, 0, 1)} CR¹ (j) X10 = {(1,2,0,0), (0, 2, 3,0), (0, 0, -1, 1)} CR¹

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