Is the binary relation defined on the set R reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (*Hint*) Is the binary relation C defined on the set P(N) reflexive? Is it symmet- ric? Is it transitive? Is it an equivalence relation? (Recall that P(N) is the set of subsets of N). Define the binary relation on C as follows: 2₁~22 iff z1 = |22|. Is ~ reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (*Hint*) Define the binary relation~ on Z as follows: a b iff |a − b < 4. Is ~ reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation?
Is the binary relation defined on the set R reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (*Hint*) Is the binary relation C defined on the set P(N) reflexive? Is it symmet- ric? Is it transitive? Is it an equivalence relation? (Recall that P(N) is the set of subsets of N). Define the binary relation on C as follows: 2₁~22 iff z1 = |22|. Is ~ reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (*Hint*) Define the binary relation~ on Z as follows: a b iff |a − b < 4. Is ~ reflexive? Is it symmetric? Is it transitive? Is it 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
Please do Exercise 17.2.14 part ABCD and please show step by step and explain
Hint A: ≤ is not symmetric – you may show this by giving a counterexample.
Hint C: The “Is it transitive?” question amounts to answering the following: Given z1 ∼ z2 and z2 ∼ z3. Is it always true that z1 ∼ z3? If yes, prove it; and if no, give a counterexample.
Transcribed Image Text:**Exercise 17.2.14.** For each of the following, explain your answers.
**17.2 PARTITIONS AND PROPERTIES OF BINARY RELATIONS**
(a) Is the binary relation \( \leq \) defined on the set \( \mathbb{R} \) reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (*Hint*)
(b) Is the binary relation \( \subseteq \) defined on the set \( \mathcal{P}(\mathbb{N}) \) reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (Recall that \( \mathcal{P}(\mathbb{N}) \) is the set of subsets of \( \mathbb{N} \)).
(c) Define the binary relation \( \sim \) on \( \mathbb{C} \) as follows: \( z_1 \sim z_2 \) iff \( |z_1| = |z_2| \). Is \( \sim \) reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (*Hint*)
(d) Define the binary relation \( \sim \) on \( \mathbb{Z} \) as follows: \( a \sim b \) iff \( |a - b| < 4 \). Is \( \sim \) reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation?
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