Please do Exercise 17.2.14 part ABCD and please show step by step and explainHint 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. **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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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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