Please do Exercise 17.2.17 part A,B, and C and please show step by step and explainHint for A: There are actually three counterexamples, you only need to find one.Hint for B: Give a specific example where a ∼ b and b ∼ c but a (/∼) c. In other words, (a, b) and (b, c) are elements of R∼, but (a, c) is not in R∼. It is not necessary for a, b, and c to be distinct. Hint for C: Explain why it is impossible to find a counterexample. Exercise 17.2.17.(a) Give a counterexample that proves that the binary relation C in Fig-ure 17.2.3 is not transitive. (*Hint*)(b) Explain why the binary relationR = {(1,4), (1, 1), (4,1)}is not transitive. (*Hint*)(c) Explain why the binary relationis transitive. (*Hint*)R = {(1,2), (1,3), (1,4)} Example 17.2.16. Let's think about binary relations on {1,2,3} as seenin Figure 17.2.3. Which of the binary relations, A, B, or C, are transitive?Why or why not?BAZA32 3C2Figure 17.2.3. Digraphs to correspond with Example 17.2.16Is the relation in A transitive? Let's consider Remember how transitivityis defined: if a ~ band b~ c then a~ c. In more prosaic terms, if there'san arrow from a to b and another arrow from b to c, then there's an arrowdirectly from a to c. We may conceptualize this as follows. Suppose a, b, andc represent airports, and arrows represent flights between airports. In termsof this example, transitivity means that whenever there's an indirect routebetween airports (with multiple stops), then there's also a direct route. Soin the case of relation A, we may notice there's an indirect route from 3 to2 by going through 1, but there's also a direct route from 3 to 2 (or moreformally, 3~ 1, 1 ~ 2, and 3~2). Furthermore this is the only examplein A of an indirect route. Therefore this relation is transitive. If 3~ 2 isremoved from this binary relation, then the relation isn't transitive becauseit's still be possible to get from 3 to 2 via 1, but there's no longer a directroute.2

Given: Digraph of figure C. To find: (a) An counterexample that proves the binary relation in…

Exercise 17.2.17. (a) Give a counterexample that proves that the binary relation C in Fig- ure 17.2.3 is not transitive. (*Hint*) (b) Explain why the binary relation R = {(1,4), (1, 1), (4,1)} is not transitive. (*Hint*) (c) Explain why the binary relation is transitive. (*Hint*) R= {(1,2), (1,3), (1,4)}

Exercise 17.2.17. (a) Give a counterexample that proves that the binary relation C in Fig- ure 17.2.3 is not transitive. (Hint) (b) Explain why the binary relation R = {(1,4), (1, 1), (4,1)} is not transitive. (Hint) (c) Explain why the binary relation is transitive. (Hint) R= {(1,2), (1,3), (1,4)}

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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