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Let A= R- {o,}. det G = {4, t, qf where 1: A> A defined by icx)> x z¢A L+ ズ-A fex) = 1ーメ+x-A 1-X 9:A A Prove that G is a A>A {-X gexs = 「ノ Subgrmp of s(A)

Let A= R- {o,}. det G = {4, t, qf where 1: A> A defined by icx)> x z¢A L+ ズ-A fex) = 1ーメ+x-A 1-X 9:A A Prove that G is a A>A {-X gexs = 「ノ Subgrmp of s(A)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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S(A)= Set of all f: A-A which is both one to one and onto.

 

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1) Let \( A = \mathbb{R} - \{0, 1\} \). Define \( G = \{ i, f, g \} \) where - \( i: A \rightarrow A \) defined by \( i(x) = x \quad \forall x \in A \) - \( f: A \rightarrow A \) defined by \( f(x) = \frac{1}{1-x} \quad \forall x \in A \) - \( g: A \rightarrow A \) defined by \( g(x) = \frac{1-x}{x} \quad \forall x \in A \) Prove that \( G \) is a subgroup of \( S(A) \).
Transcribed Image Text:1) Let \( A = \mathbb{R} - \{0, 1\} \). Define \( G = \{ i, f, g \} \) where - \( i: A \rightarrow A \) defined by \( i(x) = x \quad \forall x \in A \) - \( f: A \rightarrow A \) defined by \( f(x) = \frac{1}{1-x} \quad \forall x \in A \) - \( g: A \rightarrow A \) defined by \( g(x) = \frac{1-x}{x} \quad \forall x \in A \) Prove that \( G \) is a subgroup of \( S(A) \).
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