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Use the element method for proving that a set equals the empty set to prove the following statement. (Assume that all sets are subsets of a universal set U.) If U denotes a universal set, then UC = ø. Proof by contradiction: Consider the sentences in the following scrambled list. But, by definition of a universal set, U contains all elements under discussion, and so x E U. So, by definition of complement x E U. Let U be a universal set and suppose Uº = Ø. Let U be a universal set and suppose UC ± Ø. Then there exists an element x in UC. Thus x EU and x € U, which is a contradiction. So, by definition of complement x ¢ U. But, by definition of a universal set, UC contains no elements. We construct the proof by selecting appropriate sentences from the list and putting them in the correct order. 1. Let U be a universal set and suppose UC = Ø. 2. Let U be a universal set and suppose UC = Ø. 3. ---Select--- 4. ---Select--- 5. Thus x e U and x ¢ U, which is a contradiction. 6. Hence the supposition is false, and so UC = ø.

Use the element method for proving that a set equals the empty set to prove the following statement. (Assume that all sets are subsets of a universal set U.) If U denotes a universal set, then UC = ø. Proof by contradiction: Consider the sentences in the following scrambled list. But, by definition of a universal set, U contains all elements under discussion, and so x E U. So, by definition of complement x E U. Let U be a universal set and suppose Uº = Ø. Let U be a universal set and suppose UC ± Ø. Then there exists an element x in UC. Thus x EU and x € U, which is a contradiction. So, by definition of complement x ¢ U. But, by definition of a universal set, UC contains no elements. We construct the proof by selecting appropriate sentences from the list and putting them in the correct order. 1. Let U be a universal set and suppose UC = Ø. 2. Let U be a universal set and suppose UC = Ø. 3. ---Select--- 4. ---Select--- 5. Thus x e U and x ¢ U, which is a contradiction. 6. Hence the supposition is false, and so UC = ø.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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### Proof by Contradiction: Universal Set and Empty Set **Statement to Prove:** If \( U \) denotes a universal set, then \( U^C = \emptyset \). **Method of Proof:** The element method for proving that a set equals the empty set through contradiction. **Scrambled Sentences:** Consider the sentences in the following scrambled list: 1. But, by definition of a universal set, \( U \) contains all elements under discussion, and so \( x \in U \). 2. So, by definition of complement \( x \notin U \). 3. Let \( U \) be a universal set and suppose \( U^C = \emptyset \). 4. Let \( U \) be a universal set and suppose \( U^C \neq \emptyset \). 5. Then there exists an element \( x \in U^C \). 6. Thus \( x \in U \) and \( x \notin U \), which is a contradiction. 7. So, by definition of complement \( x \notin U \). 8. But, by definition of a universal set, \( U^C \) contains no elements. **Proof Construction:** We construct the proof by selecting appropriate sentences from the list and putting them in the correct order. 1. Let \( U \) be a universal set and suppose \( U^C \neq \emptyset \). ❌ 2. Let \( U \) be a universal set and suppose \( U^C \neq \emptyset \). ❌ 3. *---Select---* 4. *---Select---* 5. Thus \( x \in U \) and \( x \notin U \), which is a contradiction. ✅ 6. Hence the supposition is false, and so \( U^C = \emptyset \). **Explanation of Steps:** 1. **Correct Step:** Let \( U \) be a universal set and suppose \( U^C \neq \emptyset \). 2. **Correct Step:** Then there exists an element \( x \in U^C \). 3. **Correct Step:** By definition of complement \( x \notin U \). 4. **Correct Step:** But, by definition of a universal set, \( U \) contains all elements under discussion, and so \( x \in U \). 5. **Correct Step:** Thus
Transcribed Image Text:### Proof by Contradiction: Universal Set and Empty Set **Statement to Prove:** If \( U \) denotes a universal set, then \( U^C = \emptyset \). **Method of Proof:** The element method for proving that a set equals the empty set through contradiction. **Scrambled Sentences:** Consider the sentences in the following scrambled list: 1. But, by definition of a universal set, \( U \) contains all elements under discussion, and so \( x \in U \). 2. So, by definition of complement \( x \notin U \). 3. Let \( U \) be a universal set and suppose \( U^C = \emptyset \). 4. Let \( U \) be a universal set and suppose \( U^C \neq \emptyset \). 5. Then there exists an element \( x \in U^C \). 6. Thus \( x \in U \) and \( x \notin U \), which is a contradiction. 7. So, by definition of complement \( x \notin U \). 8. But, by definition of a universal set, \( U^C \) contains no elements. **Proof Construction:** We construct the proof by selecting appropriate sentences from the list and putting them in the correct order. 1. Let \( U \) be a universal set and suppose \( U^C \neq \emptyset \). ❌ 2. Let \( U \) be a universal set and suppose \( U^C \neq \emptyset \). ❌ 3. *---Select---* 4. *---Select---* 5. Thus \( x \in U \) and \( x \notin U \), which is a contradiction. ✅ 6. Hence the supposition is false, and so \( U^C = \emptyset \). **Explanation of Steps:** 1. **Correct Step:** Let \( U \) be a universal set and suppose \( U^C \neq \emptyset \). 2. **Correct Step:** Then there exists an element \( x \in U^C \). 3. **Correct Step:** By definition of complement \( x \notin U \). 4. **Correct Step:** But, by definition of a universal set, \( U \) contains all elements under discussion, and so \( x \in U \). 5. **Correct Step:** Thus
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