Question

Consider the set X=\mathbb{R}\setminus \left \{ -1 \right \}. Define * on Xby x_1*x_2=x_1+x_2+x_1x_2\forall x_1,x_2\in X.

i) Check whether (X,*) is a group or not.

ii) Prove that x ∗ x ∗ x ∗K∗ x (n times) = (1+ x) −1 ∀ n∈N n and x ∈X .

11 Mar 2024
Answer :
Word Count : 655

To solve this problem numerically, let's first define the set X and the operation * on X.

Given:
X = {x | x ≠ 1}

The operation * is defined on X by:
x * y = (xy + 1) / (x + y)

i) Checking whether (X, *) is a group or not:

For (X, *) to be a group, it must satisfy the following four group axioms:

1. Closure: For any x, y in X, x * y should also be in X.
2. Associativity: For any x, y, z in X, (x * y) * z = x * (y * z).
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