(4) Let X = {(a,b) : a,bЄ Z, aon X given by (a, b) R(c, d) if ad0,60}. Show that the relation R=bc is an equivalence relation. [1 for reflexivity, 1 for symmetry, 1 for transitivity,1 for conclusion]

To prove R is an equivalence relation, we need to show that it satisfies the three properties of an…

Answered: (4) Let X = {(a,b) : a,bЄ Z, a on X…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 12E: Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and...

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Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and justify your decisions. if and only ifand are parallel. if and only ifand are perpendicular.
Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.
In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.
2) Let Q = {0,1,2,3} and define relation R on Q as follows: R = {(0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0), (3,3)} Determine whether the R relation is reflexive, symmetric or transitive.
Let A = {1, 2, 3, 4, 5} and R be the relation defined by R = {(1,1), (2,2), (2,4), (2,5), (3,3),(4,2), (4,4)}. Justify whether relation R fulfill the property of: i) Reflexive. ii) Symmetric.(iii) Anti-symmetric.(iv) Transitive.
Let x1, x2, y1, y2 be real numbers. The expression (x1, y1) ~ (x2, y2) means that x12 + y12 = x22 + y22. Prove that the relation ~ is an equivalence relation on R2.
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3. For universe S = R2 define relation ~ as: (a, b) ~ (c,d) — 2a-b2c - d. Prove that is an equivalence relation on R2, or find a counterexample.
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ii. Determine whether the relation, R = {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)} is: Yes/No Justify your answer. Types of relations Reflexive Symmetric Antisymmetric Transitive (8 marks)

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(4) Let X = {(a,b) : a,bЄ Z, a on X given by (a, b) R(c, d) if ad 0,60}. Show that the relation R = bc is an equivalence relation. [ 1 for reflexivity, 1 for symmetry, 1 for transitivity, 1 for conclusion]