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Define a relation Q on the set Rx Ras follows. For all ardered pairs (w, x) and (y, z) in Rx R, (w, x) Q (y, z) = x-z. (a) Prove that Q is an equivalence relation. To prove that Q is an equivalence relation, It is necessary to show that Q is reflexive, symmetric, and transitive. Proof that Qis an equivalence relation: (1) Proof that Q is reflexive: Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order. By definition of Q. (W, x) - (W, x). By the reflexive property of equality, x -> By the reflexive property of equality, W - w By the symmetric property of equality, w- w. By the symmetric property of equality, x - x. Proof: 1. Suppose (w, x) is any ordered pair of real numbers. 2.-Select- 3. -Seiect 4. Hence, Q is reflexive. (2) Proof that Q is symmetric: Construct a proof by selecting sentences from the following scrambled list and putting them in the corect order. BY GGtiniton of R Y -

Define a relation Q on the set Rx Ras follows. For all ardered pairs (w, x) and (y, z) in Rx R, (w, x) Q (y, z) = x-z. (a) Prove that Q is an equivalence relation. To prove that Q is an equivalence relation, It is necessary to show that Q is reflexive, symmetric, and transitive. Proof that Qis an equivalence relation: (1) Proof that Q is reflexive: Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order. By definition of Q. (W, x) - (W, x). By the reflexive property of equality, x -> By the reflexive property of equality, W - w By the symmetric property of equality, w- w. By the symmetric property of equality, x - x. Proof: 1. Suppose (w, x) is any ordered pair of real numbers. 2.-Select- 3. -Seiect 4. Hence, Q is reflexive. (2) Proof that Q is symmetric: Construct a proof by selecting sentences from the following scrambled list and putting them in the corect order. BY GGtiniton of R Y -

Advanced Engineering Mathematics
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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(3) **Proof that Q is transitive:** **Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order:** 1. By definition of Q, \( x, y, z \rightarrow u = z \) 2. By definition of Q, \( (w, x) Q (y, z) \) 3. By definition of Q, \( (u, v) Q (w, x) \) 4. By the transitive property of equality, \( v = y \) 5. By the transitive property of equality, \( x = z \) 6. By definition of Q, \( u = w \text{ and } v = x \) **Proof:** 1. Suppose \( (w, x), (u, v), (y, z) \) are any ordered pairs of real numbers such that \( (w, x) Q (v, u) \) and \( (u, v) Q (y, z) \). 2. \[ \text{Select: } \boxed{3} \] 3. \[ \text{Select: } \boxed{2} \] 4. \[ \text{Select: } \boxed{4} \] 5. Hence, Q is transitive. (4) **Conclusion:** Since Q is reflexive, symmetric, and transitive, it is an equivalence relation. (b) **What are the distinct equivalence classes of Q?** Each equivalence class has which of the following forms? - \((x, y) | x = b \text{ for some real number } b\) - \((x, y) | x + y = a + b \text{ for some real number } b\) - \((x, y) | y = b \text{ for some real number } b\) There are as many equivalence classes as there are which of the following? (Select all that apply.) - distinct vertical lines in the plane - distinct real numbers - distinct lines in the plane whose coordinates equal each other - distinct integers - distinct horizontal lines in the plane **Need Help? Read It >**
Transcribed Image Text:(3) **Proof that Q is transitive:** **Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order:** 1. By definition of Q, \( x, y, z \rightarrow u = z \) 2. By definition of Q, \( (w, x) Q (y, z) \) 3. By definition of Q, \( (u, v) Q (w, x) \) 4. By the transitive property of equality, \( v = y \) 5. By the transitive property of equality, \( x = z \) 6. By definition of Q, \( u = w \text{ and } v = x \) **Proof:** 1. Suppose \( (w, x), (u, v), (y, z) \) are any ordered pairs of real numbers such that \( (w, x) Q (v, u) \) and \( (u, v) Q (y, z) \). 2. \[ \text{Select: } \boxed{3} \] 3. \[ \text{Select: } \boxed{2} \] 4. \[ \text{Select: } \boxed{4} \] 5. Hence, Q is transitive. (4) **Conclusion:** Since Q is reflexive, symmetric, and transitive, it is an equivalence relation. (b) **What are the distinct equivalence classes of Q?** Each equivalence class has which of the following forms? - \((x, y) | x = b \text{ for some real number } b\) - \((x, y) | x + y = a + b \text{ for some real number } b\) - \((x, y) | y = b \text{ for some real number } b\) There are as many equivalence classes as there are which of the following? (Select all that apply.) - distinct vertical lines in the plane - distinct real numbers - distinct lines in the plane whose coordinates equal each other - distinct integers - distinct horizontal lines in the plane **Need Help? Read It >**
**Definition of Relation \( Q \):** Define a relation \( Q \) on the set \( \mathbb{R} \times \mathbb{R} \) as follows: For all ordered pairs \((x, y)\) and \((y, z)\) in \( \mathbb{R} \times \mathbb{R}\), \((x, y) \, Q \, (y, z) \) if and only if \( x = z \). **Proof that \( Q \) is an Equivalence Relation:** To prove that \( Q \) is an equivalence relation, it is necessary to show that \( Q \) is reflexive, symmetric, and transitive. **(1) Proof that \( Q \) is Reflexive:** Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order: 1. By definition of \( Q \), \((x, y) \, Q \, (x, y)\). 2. By the reflexive property of equality, \( y = y \). 3. By the reflexive property of equality, \( x = x \). 4. By the symmetric property of equality, \( y = x \). 5. By the symmetric property of equality, \( x = x \). **Proof:** 1. Suppose \((x, y)\) is any ordered pair of real numbers. 2. Then, \( x = x \) by the reflexive property of equality. 3. Hence, \( (x, y) \, Q \, (x, y) \). **(2) Proof that \( Q \) is Symmetric:** Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order: 1. By definition of \( Q \), \( x = z \). 2. By definition of \( Q \), \((x, y) \, Q \, (y, z)\). 3. By definition of \( Q \), \((y, x) \, Q \, (x,z)\). 4. By the symmetric property of equality, \( x = y \). 5. By the symmetric property of equality, \( y = x \). **Proof:** 1. Suppose \((x, y)\) and \((y, z)\) are any ordered pairs of real numbers such that \((x, y
Transcribed Image Text:**Definition of Relation \( Q \):** Define a relation \( Q \) on the set \( \mathbb{R} \times \mathbb{R} \) as follows: For all ordered pairs \((x, y)\) and \((y, z)\) in \( \mathbb{R} \times \mathbb{R}\), \((x, y) \, Q \, (y, z) \) if and only if \( x = z \). **Proof that \( Q \) is an Equivalence Relation:** To prove that \( Q \) is an equivalence relation, it is necessary to show that \( Q \) is reflexive, symmetric, and transitive. **(1) Proof that \( Q \) is Reflexive:** Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order: 1. By definition of \( Q \), \((x, y) \, Q \, (x, y)\). 2. By the reflexive property of equality, \( y = y \). 3. By the reflexive property of equality, \( x = x \). 4. By the symmetric property of equality, \( y = x \). 5. By the symmetric property of equality, \( x = x \). **Proof:** 1. Suppose \((x, y)\) is any ordered pair of real numbers. 2. Then, \( x = x \) by the reflexive property of equality. 3. Hence, \( (x, y) \, Q \, (x, y) \). **(2) Proof that \( Q \) is Symmetric:** Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order: 1. By definition of \( Q \), \( x = z \). 2. By definition of \( Q \), \((x, y) \, Q \, (y, z)\). 3. By definition of \( Q \), \((y, x) \, Q \, (x,z)\). 4. By the symmetric property of equality, \( x = y \). 5. By the symmetric property of equality, \( y = x \). **Proof:** 1. Suppose \((x, y)\) and \((y, z)\) are any ordered pairs of real numbers such that \((x, y
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