Decide if the statement is true or false. Assume that D‡Ø, U‡Ø, and DCU. Ø CU This statement is
Decide if the statement is true or false. Assume that D‡Ø, U‡Ø, and DCU. Ø CU This statement is
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
![### Determine the Truth Value of the Given Statement
**Problem Statement:**
Decide if the statement is true or false. Assume that \( D \ne \emptyset \), \( U \ne \emptyset \), and \( D \subseteq U \).
\[
\emptyset \subset U
\]
**Options:**
- True
- False
In order to determine the truth value of the statement, we must consider the definition of the subset relationship (\(\subset\)). According to set theory, a set \(A\) is a subset of set \(B\) if every element of \(A\) is also an element of \(B\).
Now, in this context, the statement asks if the empty set (\(\emptyset\)) is a subset of \(U\).
### Explanation:
1. The empty set (\(\emptyset\)) is defined as a set that contains no elements.
2. By definition, the empty set is considered a subset of any set because there are no elements in the empty set that could violate the condition of being an element of another set.
Hence, for any set \(U\), it is always true that \(\emptyset \subset U\).
### Conclusion:
The statement \(\emptyset \subset U\) is **true**.
**Select "True" from the dropdown menu.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F793b6aff-f148-4523-8250-7b38a941d7e6%2F892900f8-c6bb-4676-a091-7a56e11abbf4%2Ffdvpfkk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Determine the Truth Value of the Given Statement
**Problem Statement:**
Decide if the statement is true or false. Assume that \( D \ne \emptyset \), \( U \ne \emptyset \), and \( D \subseteq U \).
\[
\emptyset \subset U
\]
**Options:**
- True
- False
In order to determine the truth value of the statement, we must consider the definition of the subset relationship (\(\subset\)). According to set theory, a set \(A\) is a subset of set \(B\) if every element of \(A\) is also an element of \(B\).
Now, in this context, the statement asks if the empty set (\(\emptyset\)) is a subset of \(U\).
### Explanation:
1. The empty set (\(\emptyset\)) is defined as a set that contains no elements.
2. By definition, the empty set is considered a subset of any set because there are no elements in the empty set that could violate the condition of being an element of another set.
Hence, for any set \(U\), it is always true that \(\emptyset \subset U\).
### Conclusion:
The statement \(\emptyset \subset U\) is **true**.
**Select "True" from the dropdown menu.**
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