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

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

how do i solve 1-4

### 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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