Let A be the set {1,3,5,7,9} and B be the set {1,2,4,8}. Let U be the set {1,2,3,4,5,6,7,8,9,10}. What are the characteristic sequences XA and XB? See P. 55 Example 2.2.3. **Example 2.2.3: Characteristic Sequences**If \( U \) is the set of the first 10 odd positive integers, \( A \) is the subset of primes in \( U \), and \( B \) is the set of multiples of 3 in \( U \), then\[\begin{align*}U &= \{ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \} & // x_i = 2i - 1. \\A &= \{ 3, 5, 7, 11, 13, 17, 19 \} \\B &= \{ 3, 9, 15 \} \\X^A &= (0, 1, 1, 1, 0, 1, 1, 0, 1, 1) \\X^B &= (0, 1, 0, 0, 1, 0, 0, 1, 0, 0).\end{align*}\]Characteristic sequences may be used as an implementation model for subsets of any given indexed set \( U \). The set operations may be done on these sequences:\[\begin{align*}X^{A \cap B}(i) &= X^A(i) \times X^B(i); \\X^{A \cup B}(i) &= X^A(i) + X^B(i) - X^A(i) \times X^B(i); \\X^{A \setminus B}(i) &= X^A(i) - X^A(i) \times X^B(i).\end{align*}\]If \( A \subseteq B \) then \( X^A(i) \leq X^B(i) \) for each index \( i \),and \[|A| = \sum_{i=1}^n X^A_i.\]

Use the example attached in the question to solve the given problem.

Answered: Let A be the set {1,3,5,7,9} and B be…

Let A be the set {1,3,5,7,9} and B be the set {1,2,4,8}. Let U be the set {1,2,3,4,5,6,7,8,9,10}. What are the characteristic sequences XA and XB? See P. 55 Example 2.2.3.

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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Question

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Let A be the set {1,3,5,7,9} and B be the set {1,2,4,8}. Let U be the set {1,2,3,4,5,6,7,8,9,10}. What are the characteristic sequences X^A and X^B? See P. 55 Example 2.2.3.

**Example 2.2.3: Characteristic Sequences**

If \( U \) is the set of the first 10 odd positive integers, \( A \) is the subset of primes in \( U \), and \( B \) is the set of multiples of 3 in \( U \), then

\[
\begin{align*}
U &= \{ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 \} & // x_i = 2i - 1. \\
A &= \{ 3, 5, 7, 11, 13, 17, 19 \} \\
B &= \{ 3, 9, 15 \} \\
X^A &= (0, 1, 1, 1, 0, 1, 1, 0, 1, 1) \\
X^B &= (0, 1, 0, 0, 1, 0, 0, 1, 0, 0).
\end{align*}
\]

Characteristic sequences may be used as an implementation model for subsets of any given indexed set \( U \). The set operations may be done on these sequences:

\[
\begin{align*}
X^{A \cap B}(i) &= X^A(i) \times X^B(i); \\
X^{A \cup B}(i) &= X^A(i) + X^B(i) - X^A(i) \times X^B(i); \\
X^{A \setminus B}(i) &= X^A(i) - X^A(i) \times X^B(i).
\end{align*}
\]

If \( A \subseteq B \) then \( X^A(i) \leq X^B(i) \) for each index \( i \),

and

\[
|A| = \sum_{i=1}^n X^A_i.
\]

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