(a) Assume that f: A onto B and g: B → A. Prove that g = f-¹ iff g of = IA or fog = IB (Theorem 4.4.4(b)).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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**(a)** Assume that \( f: A \xrightarrow{1-1 \ \text{onto}} B \) and \( g: B \rightarrow A \). Prove that \( g = f^{-1} \) if and only if \( g \circ f = I_A \) or \( f \circ g = I_B \) (Theorem 4.4.4(b)).

**(b)** Give an example of sets \( A \) and \( B \) and functions \( f \) and \( g \) such that \( f: A \rightarrow B \), \( g: B \rightarrow A \), \( g \circ f = I_A \), and \( g \neq f^{-1} \).
Transcribed Image Text:### Transcription of Educational Text **(a)** Assume that \( f: A \xrightarrow{1-1 \ \text{onto}} B \) and \( g: B \rightarrow A \). Prove that \( g = f^{-1} \) if and only if \( g \circ f = I_A \) or \( f \circ g = I_B \) (Theorem 4.4.4(b)). **(b)** Give an example of sets \( A \) and \( B \) and functions \( f \) and \( g \) such that \( f: A \rightarrow B \), \( g: B \rightarrow A \), \( g \circ f = I_A \), and \( g \neq f^{-1} \).
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