(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

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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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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