**Transcription for Educational Website** Below are statements involving set theory and real numbers. For each statement, "T" indicates that the statement is true, while "F" indicates it is false: 1. **Statement 1:** **T F** For sets A and B, suppose for all \( x \in A \) it follows that \( x \notin B \), then \( A \cap B = \varnothing \). 2. **Statement 2:** **T F** For sets A and B, if there exist \( x \in A \) and \( x \notin B \), then \( A \nsubseteq B \). 3. **Statement 3:** **T F** If \( x \) and \( y \) are real numbers and \( x^4 < y^4 \) then \( x < y \). 4. **Statement 4:** **T F** For sets A and B, \( A = (A - B) \cup (A \cap B) \). **Explanation:** - **Statement 1** involves the basic property of set intersection, stating that if no element of set A is in set B, then their intersection is the empty set. - **Statement 2** discusses the definition of subset, indicating that if there is at least one element in A not in B, then A is not a subset of B. - **Statement 3** examines properties of real numbers and powers, focusing on whether the fourth power maintains inequality. - **Statement 4** presents a property of set operations involving difference and intersection related to union. These statements and their assessment (true or false) serve as exercises in understanding fundamental concepts in set theory and real analysis.
**Transcription for Educational Website** Below are statements involving set theory and real numbers. For each statement, "T" indicates that the statement is true, while "F" indicates it is false: 1. **Statement 1:** **T F** For sets A and B, suppose for all \( x \in A \) it follows that \( x \notin B \), then \( A \cap B = \varnothing \). 2. **Statement 2:** **T F** For sets A and B, if there exist \( x \in A \) and \( x \notin B \), then \( A \nsubseteq B \). 3. **Statement 3:** **T F** If \( x \) and \( y \) are real numbers and \( x^4 < y^4 \) then \( x < y \). 4. **Statement 4:** **T F** For sets A and B, \( A = (A - B) \cup (A \cap B) \). **Explanation:** - **Statement 1** involves the basic property of set intersection, stating that if no element of set A is in set B, then their intersection is the empty set. - **Statement 2** discusses the definition of subset, indicating that if there is at least one element in A not in B, then A is not a subset of B. - **Statement 3** examines properties of real numbers and powers, focusing on whether the fourth power maintains inequality. - **Statement 4** presents a property of set operations involving difference and intersection related to union. These statements and their assessment (true or false) serve as exercises in understanding fundamental concepts in set theory and real analysis.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Transcribed Image Text:**Transcription for Educational Website**
Below are statements involving set theory and real numbers. For each statement, "T" indicates that the statement is true, while "F" indicates it is false:
1. **Statement 1:**
**T F** For sets A and B, suppose for all \( x \in A \) it follows that \( x \notin B \), then \( A \cap B = \varnothing \).
2. **Statement 2:**
**T F** For sets A and B, if there exist \( x \in A \) and \( x \notin B \), then \( A \nsubseteq B \).
3. **Statement 3:**
**T F** If \( x \) and \( y \) are real numbers and \( x^4 < y^4 \) then \( x < y \).
4. **Statement 4:**
**T F** For sets A and B, \( A = (A - B) \cup (A \cap B) \).
**Explanation:**
- **Statement 1** involves the basic property of set intersection, stating that if no element of set A is in set B, then their intersection is the empty set.
- **Statement 2** discusses the definition of subset, indicating that if there is at least one element in A not in B, then A is not a subset of B.
- **Statement 3** examines properties of real numbers and powers, focusing on whether the fourth power maintains inequality.
- **Statement 4** presents a property of set operations involving difference and intersection related to union.
These statements and their assessment (true or false) serve as exercises in understanding fundamental concepts in set theory and real analysis.
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