Define a relation Q on the set R × R as follows. For all ordered pairs (w, x) and (y, z) in R x 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:

Define a relation Q on the set R × R as follows. For all ordered pairs (w, x) and (y, z) in R x 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:

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

(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 the symmetric property of equality, z = x.
By definition of Q, w = y.
By definition of Q, x = z.
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.
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.

Define a relation Q on the set R × R as follows.
For all ordered pairs (w, x) and (y, z) in R x 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 definition of Q, (w, x) = (w, x).
By the reflexive property of equality, x = x
By the reflexive property of equality, w = w.
By the symmetric property of equality, x = x.
By the symmetric property of equality, w
= w
E W.
Proof:
1. Suppose (w, x) is any ordered pair of real numbers.
2. ---Select---
3. ---Select-.
4. Hence, Q is reflexive.

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