Given the following matrix representation of a relation R on the set A = {a, b, c, d}, which of the following tuples is not in the relation? [1 0 0 1 0 1 1 0 1 10 1 Lo 0 0 1 0
Given the following matrix representation of a relation R on the set A = {a, b, c, d}, which of the following tuples is not in the relation? [1 0 0 1 0 1 1 0 1 10 1 Lo 0 0 1 0
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
Given the following matrix representation of a relation RR on the set A={a,b,c,d}A={a,b,c,d}, which of the following tuples is not in the relation?
![**Matrix and Relations**
**Objective:**
Analyze the given matrix representation of a relation \( R \) on the set \( A = \{a, b, c, d\} \) to determine which tuple is not part of the relation.
**Matrix Representation:**
\[
\begin{bmatrix}
1 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 \\
1 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{bmatrix}
\]
**Explanation:**
This matrix is a 4x4 representation where the rows and columns correspond to the elements \( a, b, c, d \) of the set \( A \), respectively. An entry of 1 in the matrix indicates that the relation \( R \) holds between the corresponding elements, while an entry of 0 indicates it does not.
- **Row 1** (\( a \)): Related to \( a \) and \( d \).
- **Row 2** (\( b \)): Related to \( b \) and \( c \).
- **Row 3** (\( c \)): Related to \( a, b, \) and \( d \).
- **Row 4** (\( d \)): Related to \( c \).
**Question:**
Which of the following tuples is not in the relation?
- \( \circ \) \( (b, d) \)
- \( \circ \) \( (c, b) \)
- \( \circ \) \( (c, a) \)
- \( \circ \) \( (b, c) \)
**Solution:**
By examining the matrix, we can verify which tuples exist:
- \( (b, d) \) is not included, as Row 2, Column 4 is 0.
- \( (c, b) \) is included, as Row 3, Column 2 is 1.
- \( (c, a) \) is included, as Row 3, Column 1 is 1.
- \( (b, c) \) is included, as Row 2, Column 3 is 1.
Thus, the correct answer is: \( \circ \) \( (b, d) \) is not in the relation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc2e02f5-ed43-48ba-8577-f130ad68cab1%2Fda709775-338b-4a04-99ae-1bfb204059ab%2F36r9cl8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix and Relations**
**Objective:**
Analyze the given matrix representation of a relation \( R \) on the set \( A = \{a, b, c, d\} \) to determine which tuple is not part of the relation.
**Matrix Representation:**
\[
\begin{bmatrix}
1 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 \\
1 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{bmatrix}
\]
**Explanation:**
This matrix is a 4x4 representation where the rows and columns correspond to the elements \( a, b, c, d \) of the set \( A \), respectively. An entry of 1 in the matrix indicates that the relation \( R \) holds between the corresponding elements, while an entry of 0 indicates it does not.
- **Row 1** (\( a \)): Related to \( a \) and \( d \).
- **Row 2** (\( b \)): Related to \( b \) and \( c \).
- **Row 3** (\( c \)): Related to \( a, b, \) and \( d \).
- **Row 4** (\( d \)): Related to \( c \).
**Question:**
Which of the following tuples is not in the relation?
- \( \circ \) \( (b, d) \)
- \( \circ \) \( (c, b) \)
- \( \circ \) \( (c, a) \)
- \( \circ \) \( (b, c) \)
**Solution:**
By examining the matrix, we can verify which tuples exist:
- \( (b, d) \) is not included, as Row 2, Column 4 is 0.
- \( (c, b) \) is included, as Row 3, Column 2 is 1.
- \( (c, a) \) is included, as Row 3, Column 1 is 1.
- \( (b, c) \) is included, as Row 2, Column 3 is 1.
Thus, the correct answer is: \( \circ \) \( (b, d) \) is not in the relation.
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