代写MAED 5121 Algebra and Its Applications I Midterm Examination, Fall 2023代写留学生Matlab语言

日期：2025-10-30 04:59

MAED 5121 Algebra and Its Applications I

Midterm Examination, Fall 2023

1. Determine if the given relation is an equivalence relation or not. If the relation is an equivalence relation, find its equivalences classes and determine if the quotient set is a countable set or not.  Show your reasoning for full credit.

(a) ~ is the relation on R defined by a ~ b if |a − 8| = |b − 8|, for a, b ∈ R.                                  [9 points]

(b) ◇ is the relation R defined by a◇b if [a] = [b], for a, b ∈ R, where [x] denotes the largest integer not greater than x.                     [9 points]

(c) ≡ is the relation on the power set P(U) of a non-empty universal set U defined by

A ≡ B   if   (A\B) ∪ (B\A) = ∅

           for A, B ∈ P(U). [7 points]

2. Consider the cycles β = (567), σ = (1637) and τ = (1352) in the permutation group S7 .

(a) Express the permutation βσ—1τ as a product of disjoint cycles.                                         [9 points]

(b) Find the permutation (στ)2024 .                                                                                 [9 points]

(c) Determine if there exists a permutation P ∈ S7  such that PστP—1 = βσ .  Show your reasoning for full credit.                                       [7 points]

3. Let G be an abelian group with identity element e. For any two subsets A, B of G, the subset A⊙B is defined by

A ⊙ B = {xy : x ∈ A, y ∈ B} .

(a) Prove that if A, B are subgroups of G, then A ⊙ B is also a subgroup of G.                          [8 points]

(b) Prove that if A, B , C are subgroups of G such that A ⊆ B, then A ⊙ (B ∩ C) = B ∩ (A ⊙ C).        [9 points]

(c) Show by an example that A ⊙ (B ∩ C) = B ∩ (A ⊙ C) may not hold in general if A is not a subset of B.                          [8 points]

4. Prove the following results about groups.

(a) The group (R, +) of all real numbers under addition is isomorphic to the group (R+ , ·) of all positive real numbers under multiplication; i.e., one can find a bijective homomorphism from R to R+ .       [5 points]

(b) (Z, +) (i.e., the group of integers under addition) is not isomorphic to (Q, +) (i.e., the group of rational numbers under addition). Hint: Consider the solvability of certain simple equation.              [10 points]

(c) If G is a finite group of even order with identity element e, then the equation x2 = e has at least two distinct solutions.         [10 points]