Consider the relation R = {(n, m) e N × N ||n – m| < 1}. Then for example (1, 2) E Rsince |1 – 2| = 1 while (1,3) g R since |1 – 3|= 2.Prove or disprove each of the following statements:i. R is reflexive.ii. R is symmetric.iii. R is transitive.iv. R is countable.

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Answered: Consider the relation R {(n, m) e N x N…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Let R3 be the relation on Z+ such that xR3y if and only if 2x – 3y > 0. (Recall that Z+ = {1,2,3, 4, ...}, the positive integers.) 14) Prove or Disprove: R3 is anti-symmetric.
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Determine which of these relations are symmetric. The variables x, y, x', y' represent integers. O A. xRy if and only if æ + y is positive. O B. x Ry if and only if x = |y|. O C. x Ry if and only if cy is positive. OD. xRy if and only if x – y is positive. E. x Ry if and only if xy > 0.
2. Let C = D = {-3, –2, –1, 1, 2, 3} and define a relation S from C to D as follows: For all (x, y) e C x D, 1 (x, y) E S means that 1 is an integer. y a. Is 2 S 2? Is –1S – 1? Is (2, 2) e S? Is (2, –-2) e S? b. Write S as a set of ordered pairs.
(28) Let RCIX I be the relation on I = [−1, 1] given by the set pictured below. (The set contain the points on the boundary of each triangle, and all points inside these triangles.) T -1 1 Is R reflexive? Symmetric? Transitive? An equivalence relation?
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11. Let C= {O, 1, 2) and D = (2, 4, 6, 8) and define a relation R from A to B as follows: For all (x, y) E AXB, %3D %3D y+2 (x, y) e R means that is an integer. a. Is 1 R 2? Is 2 R 8? Is (1, 8) e R? Is (2, 6) e R ? b. Write R as a set of ordered pairs. c. Write the domain and co-domain of R. d. Draw an arrow diagram for R.

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Consider the relation R {(n, m) e N x N | |n – m| < 1}. Then for example (1,2) e R since |1 – 2| = 1 while (1, 3) g R since |1 – 3| = 2. %3D Prove or disprove each of the following statements: i. R is reflexive. ii. R is symmetric. iii. R is transitive. iv. R is countable.