1. For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, it is antisymmetric, and whether it is transitive. Write all the properties that may apply. a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)} c) {(2, 4), (4, 2)} d) {(1, 2), (2, 3), (3, 4)} e) {(1, 1), (2, 2), (3, 3), (4, 4)} f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}
1. For each of these relations on the set {1, 2, 3, 4}, decide whether it is reflexive, whether it is symmetric, it is antisymmetric, and whether it is transitive. Write all the properties that may apply. a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)} c) {(2, 4), (4, 2)} d) {(1, 2), (2, 3), (3, 4)} e) {(1, 1), (2, 2), (3, 3), (4, 4)} f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 21E: 21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in...
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Transcribed Image Text:1. For each of these relations on the set {1, 2, 3, 4), decide whether it is reflexive, whether it is symmetric, it is antisymmetric, and whether it is transitive. Write all the properties that
may apply.
а) {(2, 2), (2, 3), (2,4), (3, 2), (3, 3), (3, 4)}
b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}
c) {(2, 4), (4, 2)}
d) {(1, 2), (2, 3), (3, 4)}
e) {(1, 1), (2, 2). (3, 3), (4, 4)}
f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}
2. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) e R if and only if
a. x+ y.
b. ху> 1.
C. X = y + 1 or x = y- 1.
d. x is a multiple of y.
e. x and y are both negative or both nonnegative.
f. x = y?.
g. x2 y2
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