(3) Proof that Q is transitive: Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order. By definition of Q, (w, x) Q (y, z). By definition of Q, x = v and v = z. By definition of Q, w = u and u = y. By the transitive property of equality, x = Z. By definition of Q, (x, w) Q (z, y). By the transitive property of equality, w = y Proof: 1. Suppose (w, x), (u, v), and (y, z) are any ordered pairs of real numbers such that (w, x) Q (u, v) and (u, v) Q (y, z). Select--- 2. 3. 4. Select--- Select--- 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? O {(x, y) | x = b for some real number b} O {(x, y) | y= b for some real number b} O {(x, y) | x = y = b 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 integers ☐ distinct horizontal lines in the plane ☐ distinct real numbers Define a relation Q on the set R x R as follows. For all ordered 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 Q is 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 the symmetric property of equality, w = w. By the reflexive property of equality, x = x By the symmetric property of equality, x = x. By the reflexive property of equality, w = w. By definition of Q, (w, x) = (w, x). Proof: 1. Suppose (w, x) is any ordered pair of real numbers. 2. ---Select--- 3. ---Select--- 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 correct order. By definition of Q, (z, y) Q (x, w). By definition of Q, (y, z) Q (w, x). By the symmetric property of equality, y = w. By definition of Q, x = z. By the symmetric property of equality, z = x. By definition of Q, w = y. Proof: 1. Suppose (w, x) and (y, z) are any ordered pairs of real numbers such that (w, x) Q (y, z). 2. ---Select--- 3. Select--- 4. Select--- 5. Hence, Q is symmetric.

# (3) Proof that Q is transitive: Construct a proof by selecting sentences from the following scrambled list and putting them in the correct order. By definition of Q, (w, x) Q (y, z). By definition of Q, x = v and v = z. By definition of Q, w = u and u = y. By the transitive property of equality, x = Z. By definition of Q, (x, w) Q (z, y). By the transitive property of equality, w = y Proof: 1. Suppose (w, x), (u, v), and (y, z) are any ordered pairs of real numbers such that (w, x) Q (u, v) and (u, v) Q (y, z). Select--- 2. 3. 4. Select--- Select--- 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? O {(x, y) | x = b for some real number b} O {(x, y) | y= b for some real number b} O {(x, y) | x = y = b 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 integers ☐ distinct horizontal lines in the plane ☐ distinct real numbers Define a relation Q on the set R x R as follows. For all ordered 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 Q is 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 the symmetric property of equality, w = w. By the reflexive property of equality, x = x By the symmetric property of equality, x = x. By the reflexive property of equality, w = w. By definition of Q, (w, x) = (w, x). Proof: 1. Suppose (w, x) is any ordered pair of real numbers. 2. ---Select--- 3. ---Select--- 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 correct order. By definition of Q, (z, y) Q (x, w). By definition of Q, (y, z) Q (w, x). By the symmetric property of equality, y = w. By definition of Q, x = z. By the symmetric property of equality, z = x. By definition of Q, w = y. Proof: 1. Suppose (w, x) and (y, z) are any ordered pairs of real numbers such that (w, x) Q (y, z). 2. ---Select--- 3. Select--- 4. Select--- 5. Hence, Q is symmetric.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...

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