In: Advanced Math

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is complete.

Let S = {2 k : k ∈ Z}. Let R be a relation defined on Q− {0} by
x R y if x y ∈ S. Prove that R is an equivalence relation.
Determine the equivalence class

1. Let α < β be real numbers and N ∈ N.
(a). Show that if β − α > N then there are at least N
distinct integers strictly between β and α.
(b). Show that if β > α are real numbers then there is a
rational number q ∈ Q such β > q > α.
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2. Let x, y, z be real numbers.The absolute value of x is
defined by
|x|= x, if x ≥...

Let Rx denote the group of nonzero real numbers under
multiplication and let R+ denote the group of positive
real numbers under multiplication. Let H be the subgroup {1, −1} of
Rx. Prove that Rx ≈ R+ ⊕ H.

On the set S of all real numbers, define a relation R = {(a, b):a ≤ b}. Show that R is transitive.

If R is the 2×2 matrices over the real, show that R has
nontrivial left and right ideals.?
hello
could you please solve this problem with the clear hands writing to
read it please? Also the good explanation to understand the
solution is by step by step please
thank
the subject is Modern
Algebra

Let K = { s+t * 2^(1/2), such that s, t are Rational}. Show that
K is a Field

Let
R be the ring of all 2 x 2 real matrices.
A. Assume that A is an element of R such that AB=BA for all B
elements of R. Prove that A is a scalar multiple of the identity
matrix.
B. Prove that {0} and R are the only two ideals.
Hint: Use the Matrices E11, E12, E21, E22.

1) Show that if A is an open set in R and k ∈ R \ {0}, then the
set kA = {ka | a ∈ A} is open.

Let ? have the gamma (?,?) distribution.
Show that ?(?≥2?(?))≤(2/?)^(r)
[Remember that if ℎ is a positive function of ?, then the value
of ? that minimizes ℎhis the same as the value of ? that minimizes
log(ℎ)]

2. Define a relation R on pairs of real numbers as follows: (a,
b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a
partial order? Why or why not? If R is a partial order, draw a
diagram of some of its elements.
3. Define a relation R on integers as follows: mRn iff m + n is
even. Is R a partial order? Why or why not? If R is...

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