2.48. Is it true that {ax + by + cz|x, y, z ∈ Z} = {n...

2.48. Is it true that {ax + by + cz|x, y, z ∈ Z} = {n · gcd(a, b, c)|n ∈ Z}?

2.49. What are all the integer values of e for which the Diophantine equation 18x + 14y + 63z = e has an integer solution. Find a solution for each such e.

2.50. For integers a, b and k > 0, is it true that a | b iff a^k | b^k ?

Expert Solution

As per expert's policy, we are instructed to solve the first question if having no subparts (subparts like parts a,b,c,d,... or i,ii,iii,iv,....) or the first four subparts if having multiple subparts unless the student mentions specifically.

You are requested to kindly give an upvote if you satisfy the answer I have provided. Else, please leave the reason in the comment section below. Your feedback is highly valuable. You are much appreciated. Thank you.

Colby Messinger answered 9 months ago

For each of the formulas below, state whether it is true or false. a) pX,Y,Z(x,y,z)=pY(y)pZ∣Y(z∣y)pX∣Y,Z(x∣y,z) ...

For each of the formulas below, state whether it is true or false. a) pX,Y,Z(x,y,z)=pY(y)pZ∣Y(z∣y)pX∣Y,Z(x∣y,z) Select an option True False b) pX,Y∣Z(x,y∣z)=pX(x)pY∣Z(y∣z) Select an option True False c) pX,Y∣Z(x,y∣z)=pX∣Z(x∣z)pY∣X,Z(y∣x,z) Select an option True False d) ∑xpX,Y∣Z(x,y∣z)=1 Select an option True False e) ∑x∑ypX,Y∣Z(x,y∣z)=1 Select an option True ...

Suppose that X, Y and Z are normally distributed where X ≈ N(100,100), Y ≈ N(400,...

Suppose that X, Y and Z are normally distributed where X ≈ N(100,100), Y ≈ N(400, 400) and Z ≈ N(64,64). Let W = X + Y + Z. a) Describe the distribution of W, give a name and parameters E(W) and Var(W). b) Use Excel or R to generate 200 random values for X, Y and Z. Add these to obtain 200 values for W. Create a histogram for W. In Excel use the NORMINV(rand(),mean, sd) function. c) Estimate...

4. The product y = Ax of an m n matrix A times a vector x...

4. The product y = Ax of an m n matrix A times a vector x = (x1; x2; : : : ; xn)T can be computed row-wise as y = [A(1,:)*x; A(2,:)*x; ... ;A(m,:)*x]; that is y(1) = A(1,:)*x y(2) = A(2,:)*x ... y(m) = A(m,:)*x Write a function M-file that takes as input a matrix A and a vector x, and as output gives the product y = Ax by row, as denoted above (Hint: use a for...

10. Suppose that X, Y and Z are normally distributed where X ≈ N(100,100), Y ≈...

10. Suppose that X, Y and Z are normally distributed where X ≈ N(100,100), Y ≈ N(400, 400) and Z ≈ N(64,64). Let W = X + Y + Z. a) Describe the distribution of W, give a name and parameters E(W) and Var(W). b) Use Excel or R to generate 200 random values for X, Y and Z. Add these to obtain 200 values for W. Create a histogram for W. In Excel use the NORMINV(rand(),mean, sd) function. c)...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just...

The curried version of let f (x,y,z) = (x,(y,z)) is let f (x,(y,z)) = (x,(y,z)) Just f (because f is already curried) let f x y z = (x,(y,z)) let f x y z = x (y z)

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}....

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}. a) Prove or disprove: A ⊆ X b) Prove or disprove: X ⊆ A c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y ) d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

If X, Y and Z are three arbitrary vectors, prove these identities: a. (X×Y).Z = X.(Y×Z)...

If X, Y and Z are three arbitrary vectors, prove these identities: a. (X×Y).Z = X.(Y×Z) b. X×(Y×Z) = (X.Z)Y – (X.Y)Z c. X.(Y×Z) = -Y.(X×Z)

Write script in Matlab to solve 1. Y=Ax where X is of dimension n by 1...

Write script in Matlab to solve 1. Y=Ax where X is of dimension n by 1 and A is of dimension m by n. 2. C= A*B where A is of dimension m by p and B is dimension of p by n. So write matlab script to obtain Y and C in above problems.

Given the function u(p,q,r)=((p-q)/(q-r)), with p=x+y+z,q=x-y+z, and r=x+y-z, find the partial derivatives au/ax=, au/ay=, au/az=

Digital Logic Simplify: F = (x’∙ y’∙ z’) + (x’∙ y ∙ z’) + (x ∙...

Digital Logic Simplify: F = (x’∙ y’∙ z’) + (x’∙ y ∙ z’) + (x ∙ y’ ∙ z’) + (x ∙ y ∙ z) F = (x + y + z’) (x + y’ + z’) (x’ + y + z’) (x’ + y’ + z)

Question