I need help on this dicrete mathematics problem where I must define a relation R.5. Define a relation < on the set of ordered pairs of real numbers (x, y) as follows:(x1,)<(x2, y2) if and only if (1) x, < x, or (2) x, = x, & y, < y,. Prove that < defines a linearordering on the set of ordered pairs of real numbers. (This shows that the coordinate plane canbe linearly ordered!)

The relation ≺ is defined as follows: x1,y1≺x2,y2 if and only if x1<x2 or x1=x2 and y1≤y2

R.5. Define a relation < on the set of ordered pairs of real numbers (x, y) as follows: (X:V1)< (x2, V2) if and only if (1) x < x, or (2) x, = x, & y, < y2. Prove that < defines a linear ordering on the set of ordered pairs of real numbers. (This shows that the coordinate plane can be linearly ordered!)

R.5. Define a relation < on the set of ordered pairs of real numbers (x, y) as follows: (X:V1)< (x2, V2) if and only if (1) x < x, or (2) x, = x, & y, < y2. Prove that < defines a linear ordering on the set of ordered pairs of real numbers. (This shows that the coordinate plane can be linearly ordered!)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 2E: 2. In each of the following parts, a relation is defined on the set of all integers. Determine in...

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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. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric or transitive. Justify your answers. a. if and only if . b. if and only if . c. if and only if for some in . d. if and only if . e. if and only if . f. if and only if . g. if and only if . h. if and only if . i. if and only if . j. if and only if . k. if and only if .
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
3. For each of the following mappings, write out and for the given and, where.
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.