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4. Prove that, if f : A → B has an inverse, then f is bijective. (The converse is also true, and very useful, but you do not need to prove it.)

4. Prove that, if f : A → B has an inverse, then f is bijective. (The converse is also true, and very useful, but you do not need to prove it.)

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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### Bijective Functions and Binary Operations in Algebra #### Bijective Functions **Theorem:** Prove that, if \( f : A \rightarrow B \) has an inverse, then \( f \) is bijective. (The converse is also true, and very useful, but you do not need to prove it.) #### Binary Operations If \( A \) is a set, a **binary operation** on \( A \) is a function \( \beta : A \times A \rightarrow A \). Binary operations are ubiquitous in modern algebra, and their appearance there motivates the following notation: For \((a_1, a_2) \in A \times A\), we write \( a_1 \beta a_2 \) instead of the usual functional notation \( \beta(a_1, a_2) \) for the image of \( (a_1, a_2) \) under \( \beta \). The most important (and motivating) instances of this (very general) notion for us are the addition and multiplication operations on a ring.
Transcribed Image Text:### Bijective Functions and Binary Operations in Algebra #### Bijective Functions **Theorem:** Prove that, if \( f : A \rightarrow B \) has an inverse, then \( f \) is bijective. (The converse is also true, and very useful, but you do not need to prove it.) #### Binary Operations If \( A \) is a set, a **binary operation** on \( A \) is a function \( \beta : A \times A \rightarrow A \). Binary operations are ubiquitous in modern algebra, and their appearance there motivates the following notation: For \((a_1, a_2) \in A \times A\), we write \( a_1 \beta a_2 \) instead of the usual functional notation \( \beta(a_1, a_2) \) for the image of \( (a_1, a_2) \) under \( \beta \). The most important (and motivating) instances of this (very general) notion for us are the addition and multiplication operations on a ring.
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