Certainly! Here's a transcription suitable for an educational website:---**Exercise on Relations****4. Define a relation \( R \) on \( \mathbb{N} \) by \( aRb \) if \( a \) and \( b \) have no common factors except 1.**(a) **(5 points)** Give an example of a pair \((a, b)\) with \((a, b) \in R\).(b) **(5 points)** Give an example of a pair \((a, b)\) with \((a, b) \notin R\).---The question refers to a mathematical relation where two numbers are related if they are coprime (i.e., their greatest common divisor is 1). The task involves providing examples of number pairs that are coprime and those that are not.

Answered: 4. Define a relation R on N by aRb if a…

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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5. Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs (1, 1), (1, 2), (1, 3), (2, 3), (2, 4), (3, 1), (3, 4), (3, 5), (4, 2), (4, 5), (5, 1), (5,2), and (5, 4). Find, a) R² b) R³ c) R4 d) R5
Let X = {1, 2, 3, 4}. Let R = {(1, 3), (2, 1), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (a, b) }, where a and b are elements of X. If R is a transitive relation on X, what is the pair (a, b)? Answer:
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4. Let R be the relation defined by R= {(0, 1), (0, 2), (2,0), (2, 3), (4, 4), (2, 4), (3, 3), (1, 2), (1, 3), (3, 2)}. Evaluate dom(R), ran(R), Ro R, R|{1}, R-1 | {1} , R[{1}], and R-1[{1}].
2. Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) eR if the integer (a- b) is divisible by 4. List the elements of R and its inverse? b) Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where R = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric, transitive? Justify your answer in each case? c) Let S = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c), (c, b)}, find [a], [b] and [c] (that is the equivalent class of a, b, and c). Hence or otherwise find the set of equivalent class of a, b and c? 3. a. Define the following with an example; i. paths ii. simple graph ro 3 3 0 0 1 1 L2 1 2 0. b.Draw the graph with the adjacency matrix with respect to the 2 ordering of vertices, a, b, c, d. i. Find the degree of each vertex in your graph from part (a) above. ii. How many walks of length 2 are there from the vertex c to c? How many of these…
1. (1 Dints) Let A = {a,b,..., z} and B = {1, 2,..., 10} and define f = {(c, 6), (b, 9), (b, 8), (d, 10), (c, 10), (a, 1)} (a) Is f a relation from A to B? Why or why not? view nislopes ynin (s) odno Jon at ydw nislqxs ylsi18 (d) are there from A to B Brie (b) Explain why f is not a function from A to B. fe to squer adi at jedW (o)
Let S = {a, b, c}. Then R = {(a, b)} is a relation on S. Which of the properties reflexive, symmetric andtransitive does the relation R possess? the relation is not reflexive or symmetric but it is vacuously transitive. What I dont understand is why is it not vacuously reflexive or symmetric, explain step by step please
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4. Define the relation r on S = {A, B, C, D} by r ={ (A,B), A,D), (B,A), (B,C), (C,A), (C,B), (D,B)}. Give the relation digraph for r.
a. Write R as a set of ordered pairs.
b. Write V as a set of ordered pairs.
3. Let S = R × R and R be the relation on S defined by (a, b)R(c, d) iff a+c=b+d. Determine whether R is reflexive, symmetric, and/or transitive. %3D

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4. Define a relation R on N by aRb if a and b have no common factors except 1. (a) Give an example of a pair (a, b) with (a, b) € R. (b) Give an example of a pair (a, b) with (a,b) & R.