Tutorial Sheet 3: Algebraic Structures

 

 

Exercise 1

Let * be an internal composition law on ℝ \ {2} defined by:

x * y = xy - 2x - 2y + 6.

  1. Is the law * commutative? associative?
  2. Show that the law * has a neutral element.
  3. Determine the inverse of an element x in ℝ \ {2}.
  4. What is your conclusion?
  5. Solve in ℝ \ {2} the following equation: x * 4 = 5x + 1.
  6. For every integer n ≥ 2 and for x ∈ ℝ \ {2}, show that:

x * x * … * x (n times) = (x - 2)^n + 2.

Exercise 2

  1. Let ⋆ be an internal composition law on ]-1, 1[ defined by:

x ⋆ y = (x + y) / (1 + xy).

Show that (]-1, 1[, ⋆) is an abelian group.

  1. Consider the group (ℝ, +) and let f: ]-1, 1[ → ℝ be a map defined by:

f(x) = ln((1 - x) / (1 + x)).

Show that f is a group homomorphism.

Last modified: Friday, 5 September 2025, 1:10 PM