Tutorial Sheet 3: Algebraic Structures

 

 

Exercise 5

Let (G, *) be a group and let f be a mapping defined by:

f : G → G, x ↦ f(x) = x^(-1), where x^(-1) is the inverse of x with respect to the operation *.

  1. Show that ∀x, y ∈ G, (x * y)^(-1) = y^(-1) * x^(-1).
  2. Show that f is a homomorphism if and only if the operation * is commutative.

Exercise 6

Let ⋆ and ⋄ be two internal composition laws on ℝ defined by:

x ⋆ y = x + y - 1 and x ⋄ y = x + y - xy.

  1. Show that (ℝ, ⋆, ⋄) is a commutative ring with unity.
  2. Is (ℝ, ⋆, ⋄) a field?
آخر تعديل: الجمعة، 5 سبتمبر 2025، 1:22 PM