Show that ∗ : R × R → R defined by        a ∗ b = a + 2b is not commutative.

Q: Let G= Z* and let * be an operation in a * b. Is a * b = ab/2 is closed under the operation?

A: Disproof by an example.

Q: Prove that if A is invertible and AB = 0, then B= 0. Give a counterexample to show that the result…

A:

Q: Prove that the dual of (R², || · ||,) is isometrically isomorphic to (R², || · .).

A:

Q: 2.4 (Ross, 2003, page 593) Show that the relationship $ (x) = x;¢(1;, x) + (1 – x;)ø(0¡, x) is true…

A:

Q: Prove that (C, +) ≈ (R × R, +).

A:

Q: If B is invertible, prove that det(B-'AB) = det(A).

A:

Q: Prove that K = {x+yv3 ] x, y E Z} is a subring of R.

A:

Q: If A²=0, prove that A cannot be invertible.

A: Given,If A²=0, prove that A cannot be invertible.

Q: Exercise 8.5. (a) Is Z₁ Z6 isomorphic to Z24? (b) Is Z3 Z7 isomorphic to Z21? (c) Is Z10 Z20…

A: 8.5 (a) Is ℤ4⊕ℤ6 is isomorphic to ℤ24? (b) Is ℤ3⊕ℤ7 is isomorphic to ℤ21? (c) Is ℤ10⊕ℤ20 is…

Q: Prove that S4 is not isomorphic to D12.

A:

Q: o Show that X2 = {(x, y) E X x X : x<y} is a sublattice of X x X. %3D

A: Given that X is a lattice. To show that X[2]  is a sub lattice of X×X. It is enough to show that,…

Q: Prove that c = cn. i=1

A:

Q: Given f(x, y, z) = xV/x² - yz - - Determine of дz . O of дz of дz af дz of дz = = x² - у 2√x² - yz…

A: Find the derivative

Q: Prove that ifx > 3 and y> 4, then xy > 11.

A:

Q: 2. Prove that if AB = AC, then B need not equal C unless A is invertible.

A:

Q: Prove that if A is invertible and BA = CA, then B = C.Give a counterexample to show that the result…

A: Given: A is invertible and BA = CA  To prove: B = C

Q: Prove that R is not reflexive.

A:

Q: © If g(x, y) = In then xgx - vgw in simplified form is [c].

A: The objective is to solve partial derivative problem and explain process.

Q: If the i, j entry of A is i times j, show that det A = 0. (Exception when A = [ 1 ] .)

A:

Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: 3. prove that: det (AB) = det (A) det (B) a - (; ), » - ( ) 2 -1 7 2 B = 2 в 3 %3D

A:

Q: Prove that for a E N, a always invertible in Z,2+3a+1·

A: We know that an element a∈ℤn is called invertible if there is an element b∈ℤn such that a+b≡0  mod…

Q: Let T={a: a ER}, then T is Borel algebra.

A: Borel sets of real numbers are definable as follows. Given some set, S, a σ-algebra over S is a…

Q: Prove that if A is invertible and AB = O, then B = O.

A: To prove that if A is invertible and AB=0, then B=0.

Q: Exercise 2. Prove that if A = B and C z D, then A × C = B x D.

A: Given data: A≈B......(1)C≈D......(2) The objective is to prove A×C=B×D

Q: (b). Let Y = {a, e, o, i, u}, take two relations R and S on Y and Prove that (SoR)-1 : R-1os-1.

A: composition of relations

Q: If A is diagonalizable, then A is similar to D. Select one: O True O False

A: Given   if matrix a is diagonalizable then A is SIMILAR  to D

Q: If Σa, is divergent and Eb, is divergent, then Z(a + b) is divergent.

A: The given statement is false as sum of two divergent series may converge. We illustrate this as…

Q: Prove that if A ϵ Mnxn(F) has two identical columns, then det(A) = 0.

A: Given: A∈Mn×nF i.e. A is a matrix of order n×n.   Consider a matrix with first two column as…

Q: Let T={[a,0) : aER}, then T is not Borel algebra

A: Given a statement

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: Let x, y, A, u E Z. Prove that if 1 = Ax + µy then r, y are relatively prime.

A:

Q: Prove that if a, b E Z - {0}, then gcd(a, b) = gcd(a, -b)

A: soln:gcda,b=Greatest common divisor of a and bEvery integer that divides a and bsimultaneously will…

Q: Let -{[:);}--{:)[:} -{{)}} Both B and C are bases for R2. If {x}s =: -[:], what is {x}c?

A:

Q: If Ea, is divergent and 2b, is divergent, then E(a, + ba) is divergent.

A: Taking an example and than we are trying to understand tha statement.

Q: Prove A = BH (x)A = (Vx)B without using the equivalence theorem.

A:

Q: Given: mAB = x Prove: mz1 = 180° - x A D Exercises 25, 26

A: Given:           m AB⏜ = x   To prove:              m&lt;1 = 1800 - x

Q: The identity element of binary operation defined on ℝ by x * y= xy  is

A:

Q: Prove that ||ư + B|| = ||T|| + ||B|| if and only if i = kủ. %3D

A:

Q: Let R→R be defined by g(x) 2y+1 2y+2 Is g bijective? Prove or give a counter example.

A:

Q: 1. Let 1, y € R" be given. (a) If ||z|| = ||y|| = 1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1…

A: \documentclass{article}\usepackage{amsmath,amssymb}\begin{document} (a) Norm x=Norm y=1\\ Let angle…

Q: Suppose A and B are invertible then det(A) = det(B) %3D O True O False

A: Consider A and B are invertible

Q: If X ∼ N (μX , σ2X) and Y ∼ N (μY , σ2Y ) are independent, then prove

A: X~Nμx,σ2x and Y~Nμy,σ2y are independant

Q: prove that for operatrs a and b which comute with thier commuters[a,b]=c,so that [a,c}=[b,c}=0 ,…

A: We know that ekABe-kA=B+kA,B+k22!A,A,B+... Given that A and B commute with their commutator then…

Q: Prove that B (xX+1; y ) + B (x,y+1)=B (X,Y) let П (х+1) = ХГ (Х)

A: We have the formula for beta function interms of gamma function is that B(m,n)=Γ(m)Γ(n)Γ(m+n)As we…

Q: Let 9-(a: a€R}, then T is Borel algebra.

A: Clearly the measure of the set is =0 .

Q: A subset W of R° is given by = {(x,y, z) | 2x = y = 2z}. (i) Write down the distinct elements of W.

A:

Q: Let A EMn (R) be given. If 11x11 = 11yll = 1 and x •y = 1, show that x=y

A:

Q: Prove that if A is invertible and AB=0, then B=0.

A: Given:

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.