1. define For R", one can consider the following norm structures. For all x = (1,,n) R", we • == |||||= max {1,, n} (a) Prove that (R", || ||1) and (R", || ||∞) satisfy the triangle inequality. (b) Draw figures for S₁ = {r R² ||||₁ = 1} and S = {x € R² : ||r||∞ = 1}. ∞

(a) Let M R. Give the radius r and the center c of B(-2,5) n B(6, 7). (b) Let M = R. Using interval notations, give a simplified expression for the set L= [B(1,1) UB(5,1)] n B(3,2). (c) Let = 3(-1)" + and A = {,: ne N). Give the set W of accumulation points of A. (d) Let A₁-(-2n, 2n) and K = U An. Give a simplified expression for the interior Kº of K. NEN (e) Give the set T of isolated points of B(0, 1) U (3) U (2,5,7) (f) Let [P,Q] be a segment in R2 with midpoint H, and let (1,0) and (2, 1) be the components of the points P and H, respectively. Give the components (x, y) of the point Q. (g) Suppose that f: (M, d) →R satisfies d(f(x), f(y)) - vl. Let 21 e M and set n+1 = f(n), for n e N". Assume that M is complete and let a = lim zn. Give a simple formula satisfied by a.

C) If an expression is given that A+A'B'C=A+B'C, find the minimal expression of the function F = A+A'B'C+BC? B'+C O A+C AC +B' OA'+B D) F = (A+BC')(AB'+C) O A+B+C' OAC'+ B A(B'+ C) B+ AC

show ur work plz!

let U and v be bounded sets. if U≤Viten M+ (V) ≤M₂ (V) and M^ (U) SM" (V) proof we we prove the first inequality leaving (20)

Find the cross product axb a=(2, – 1, 2), b=<0, 4, 5) |

Setx = [ 0 : 4, 4,−4, 1, 1]' and y = ones(9, 1) Use the MATLAB function norm to computethe values of ||x||, ||y||, ||x + y|| and to verifythat the triangle inequality holds. UseMATLABalso to verify that the parallelogram law||x + y||2 +||x − y||2 = 2(||x||2 + ||y||2)

Let φ1(x) = 1, φ2(x) = x/|x|, φ3(x) = x^2 − 1/3 for −1 ≤x ≤1.Show that (φ1,φ2,φ3) form an orthogonal set.

Q1) Choose the correct answer 1- Let X be a non-empty set and r = (-0,a)| a € R) URU(4) then r is called: (11) Indiscrete (1) Discrete (I) Left ray (IV) Right ray 2- Let X be a topological space and AS X, then the largest open set contained in A is: (1) Ext (A). (1) A". (IV) Ā (1) A'. 3- Two subsets A and B of the space X are called pair of disconnection of X if (1) AnB = 6, AUB = x () AnB + (m) AUB = X 4- Let X = (a, b, c), and 1= (X, , [a), (b,e}}, then the subset {c) is : (1) open and closed. (IV) All. (1) open. (I) closed. (IV) not open not closed Q2) True or false (1) B = {[a), (b), (c} is a base of the discrete topology on X = (a, b,c). (2) r= {x,. (x,y). (x,z). [y.z)} is not a topology on X = {x, y,z}. (3) For X = (a, b, c), the excluding point topology is E, = {X, g. (a), (c). (a, c), (a. b}). (4) For a discrete topology on X and o + ACX, the cdosure of A is A. (5) For an indiscrete topology on X = {a, b, c) and A = {a, b), the limit points of A is (a, b). (6) For the usual…