1. (i) Let AC R. The characteristic function of the set A is defined to be Xa(x) = 1 if x € A and XA(z) = 0 if r is not in A. What is x[1,21 (3)? (ii) Is x1.21 1-1 from R to R? Why? Is it surjective? Why? (iii) Show that X[-1,0]U[1,2| = X[-1,0] † X[1,2]|
1. (i) Let AC R. The characteristic function of the set A is defined to be Xa(x) = 1 if x € A and XA(z) = 0 if r is not in A. What is x[1,21 (3)? (ii) Is x1.21 1-1 from R to R? Why? Is it surjective? Why? (iii) Show that X[-1,0]U[1,2| = X[-1,0] † X[1,2]|
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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![1. (i) Let \( A \subset R \). The characteristic function of the set \( A \) is defined to be \( \chi_A(x) = 1 \) if \( x \in A \) and \( \chi_A(x) = 0 \) if \( x \) is not in \( A \). What is \( \chi_{[1,2]}(3) \)?
(ii) Is \( \chi_{[1,2]} \) 1-1 from \( R \) to \( R \)? Why? Is it surjective? Why?
(iii) Show that \( \chi_{[−1,0] \cup [1,2]} = \chi_{[−1,0]} + \chi_{[1,2]} \).
(iv) Does \( \chi_{[−1,1] \cup [0,4]} = \chi_{[−1,1]} + \chi_{[0,4]} \)? Does \( \chi_{[−1,1] \cup [0,4]} = \chi_{[−1,1]} + \chi_{[0,4]} − \chi_{[−1,1] \cap [0,4]} \)? Prove your answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F6b569000-9a58-4a3e-822b-e9318e9b4285%2F70xfl2a_processed.png&w=3840&q=75)
Transcribed Image Text:1. (i) Let \( A \subset R \). The characteristic function of the set \( A \) is defined to be \( \chi_A(x) = 1 \) if \( x \in A \) and \( \chi_A(x) = 0 \) if \( x \) is not in \( A \). What is \( \chi_{[1,2]}(3) \)?
(ii) Is \( \chi_{[1,2]} \) 1-1 from \( R \) to \( R \)? Why? Is it surjective? Why?
(iii) Show that \( \chi_{[−1,0] \cup [1,2]} = \chi_{[−1,0]} + \chi_{[1,2]} \).
(iv) Does \( \chi_{[−1,1] \cup [0,4]} = \chi_{[−1,1]} + \chi_{[0,4]} \)? Does \( \chi_{[−1,1] \cup [0,4]} = \chi_{[−1,1]} + \chi_{[0,4]} − \chi_{[−1,1] \cap [0,4]} \)? Prove your answers.
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