### Compactness with two equivalent norms

(See attached file for full problem description and symbols) --- Assume that and are two equivalent norms on X, and that . Prove that M is compact in if and only if M is compact in .

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(See attached file for full problem description and symbols) --- Assume that and are two equivalent norms on X, and that . Prove that M is compact in if and only if M is compact in .

(See attached file for full problem description with symbols) --- We have seen that the linear operator defined by is represented in the standard ordered basis by the matrix . This operator satisfies . Prove that if S is a linear operator on such that , then S = 0 or S = I, or these is an ordered basis for such that

(See attached file for full problem description) --- Let V be a two-dimensional vector space over the field F, and let be an ordered basis for V. If is a linear operator and then prove that ---

6. Students at an elementary school tried an experiment. When recess was over, each student walked into the school one at a time. The first student opened all the first 100 locker doors. The second students closed all the locker doors with even numbers. The third student changed all the locker doors with numbers that were mul

(See attached file for full problem description) --- Given positive integers a and b, the extended Euclidian algorithm constructs sequences qn, rn, sn and tn, which are defined recursively as follows: q0=0, q1=0, qn= q└ rn-2/ rn-1 ┘ for n>=2; r0=a, r1=b, rn= rn-2 - qnrn-1 for n>=2; s0=1, s1=0, sn= sn-

Please help with the following proofs. Answer true or false for each along with step by step proofs. 1) Prove that all integers a,b,p, with p>0 and q>0 that ((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q Or give a counterexample 2) prove for all integers a,b,p,q with p>0 and q>0 that ((a-b)mod p) mod q=0

(See attached file for full problem description) --- Let d,m and n be positive integers with m>1 and m≡ 1 (mod d), let n= c0+mc1+m2c2+m3c3+...+mrcr be the base=m expansion of n, and let f = c0+c1+c2+c3+...+cr Prove that n is divisible by d if and only if f is divisible by d. ---

Prove for all positives integers x and y that Lcm(5x,7y) = 5* 7 * x*y ----------------------- gcd(x*gcd(5,y),7y)

How do I compute the addition and multiplication tables for: Z mod 2 [x] / (1+x^2)? z mod 2 [x] = 1 z mod 2 [x] = 0 z mod 2 [x] = 1+x z mod 2 [x] = 1+ x + x^2 I think that 1+ x^2 = 0 so x^2 = -1. Thats as far as I got.

How do I proof of a Cantor ternary set and how to identify whether its countable or not? (See attached file for full problem description with equation) --- Consider the set C all elements of R that have the form Where each αi is either 0 or 2. Prove that in fact S is the Cantor ternary set. Given that C is the C

I would like to know how to prove sets involving intersection/union. (See attached file for full problem description)

(See attached file for full problem description with proper symbols) --- Premise : (Ex) (Ey) (Axy v Bxy)  (Ez) (Cz) Premise : (x) (y) (Cx  ~Cy) Conclusion : /. : (x)(y)(~Axy) The E's should be backwards. I couldn't find a symbol for it. The /.: means concludes This symbol  means implies. T

Modern Algebra Set Theory (XVII) Laws of Algebra of Sets

Modern Algebra Set Theory (XI) Laws of Algebra of Sets

Modern Algebra Set Theory (IX) Laws of Algebra of Sets De

6. Let (G, *) be a group. Show that each equation of either the form ax = b or the form xa = b has a unique solution in G. 7. Show that (R - {1}, *), where a * b = a + b + ab is a group

For the s y m m e t r i c g e n e r a l i z e d e i g e n p r o b l e m A y = B y , l e t ' s p r o v e t h e o r t h o g o n a l i t y s t a t e m e n t s y JT B y i = 0 , J≠ i a n d yJT A y i = 0 , J ≠ i f o r J = i . F i r s t , w r i

22. Simplify these expressions (+ = + inside a circle or 'direct sum') 1. x + 0 2. x + 1 3. x + x 4. x + x-bar For each topic, demonstrate a knowledge and capability by giving the following information: 1) Problem Solution: (solution for an even number problem) 2) Personal Observation: (personal comment

Find a transitive closure of the relation R on {a,b,c,d,e} given by R= {(a,b), (a,c), (a,e),(b,a), (b,c),(c,a), (c,b),(d,a,),(e,d)}

Show that the symmetric closure of the union of 2 relations is the union of their symmetric closures.

6. a) What does it mean for a set of operators to be functionally complete? b) Is the set {+, .} functionally complete? c) Are there sets of single operator that are functionally complete? Please see the attached file for the fully formatted problems.

Modern Algebra Set Theory (VIII) Laws of Algebra of Sets

Designate each simple statement with a letter. Then write down the compound statements using the following rules(modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, addition, simplification, or resolution), arrive at the conclusion. I've done all but the latter. If you send an email then I will write t

Modern Algebra Set Theory (V) Laws of Algebra of Sets

Modern Algebra Set Theory (III) Equivalence Classes of an Equivalence Relation Let ~ be an equi

Modern Algebra Set Theory (II) Equivalence Classes of an Equivalence Relation The distinct e

Modern Algebra Set Theory (I) Equivalence Relation Let S be the set of all i

A) Let n and r be positive integers. Explain why the number of solutions of the equation x1 + x2 + ... + xn = r, where xi is a nonnegative integer for i = 1, 2, 3, ..., n, equals the number of r-combinations of a set with n elements. b) How many solutions in nonnegative integers are there to the equation x1 + x2 + x3 + x4 = 17

22. Prove or disprove then n^2 - 1 is composite whenever n is a positive integer greater than 1.

(See attached file for full problem description with proper symbols and equations) --- 1)Prove that for any non-empty sets A x (B-C) = (AxB)-(AxC) 2) Let a,b be integers and m a positive integer. Prove that: ab = [(a mod m ) * (b mod m) mod m ] 3)Prove or disprove (a mod m) + (b mod m) = (a+b) mod m for all intege