Suppose that A is a set, and fC Ax A is a relation on A such that f is symmetric f is a function Prove that f is a bijection of A with itself, that is, prove that f is both injective and surjective. (Hint: recall that a relation f C Ax A is a function iff (i) (Va e A)(3b € A)[(a, b) E f] (ii) if (a, b) E ƒ ^ (a, c) e f, then b= c.)
Suppose that A is a set, and fC Ax A is a relation on A such that f is symmetric f is a function Prove that f is a bijection of A with itself, that is, prove that f is both injective and surjective. (Hint: recall that a relation f C Ax A is a function iff (i) (Va e A)(3b € A)[(a, b) E f] (ii) if (a, b) E ƒ ^ (a, c) e f, then b= c.)
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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![8.
Suppose that A is a set, and f CAx A is a relation on A such that
- f is symmetric
- f is a function
Prove that f is a bijection of A with itself, that is, prove that f is both injective and surjective.
(Hint: recall that a relation f Ç Ax A is a function iff (i) (Va E A)(3b E A)[(a,b) E f] (ii) if
(a, b) E ƒ ^ (a, c) € f, then b= c.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F039207b8-632a-4dc4-a3c4-fd77b3c684b4%2F5e112194-023e-41b0-9a9a-1168e83634fd%2Fcmv0mqq_processed.png&w=3840&q=75)
Transcribed Image Text:8.
Suppose that A is a set, and f CAx A is a relation on A such that
- f is symmetric
- f is a function
Prove that f is a bijection of A with itself, that is, prove that f is both injective and surjective.
(Hint: recall that a relation f Ç Ax A is a function iff (i) (Va E A)(3b E A)[(a,b) E f] (ii) if
(a, b) E ƒ ^ (a, c) € f, then b= c.)
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