Question

A certain relation * is defined on the set $mathbb{R}^2$ by $(x_1, x_2) * (y_1, y_2)$ if and only if there exists a real number $0 < k le 1$ such that $y_2 = kx_1$ and $x_2 = ky_1$. For each of the following questions, be sure to provide a proof supporting your answer. a) Is * reflexive? b) Is * symmetric? c) Is * anti-symmetric? d) Is * transitive? e) Is * an equivalence relation, a partial order, both, or neither?

          A certain relation * is defined on the set $mathbb{R}^2$ by
$(x_1, x_2) * (y_1, y_2)$ if and only if there exists a real number $0 < k le 1$ such that
$y_2 = kx_1$ and $x_2 = ky_1$.
For each of the following questions, be sure to provide a proof supporting your answer.
a) Is * reflexive?
b) Is * symmetric?
c) Is * anti-symmetric?
d) Is * transitive?
e) Is * an equivalence relation, a partial order, both, or neither?

A certain relation * is defined on the set mathbbR^2 by
(x1, x2) * (y1, y2) if and only if there exists a real number 0 < k le 1 such that
y2 = kx1 and x2 = ky1.
For each of the following questions, be sure to provide a proof supporting your answer.
a) Is * reflexive?
b) Is * symmetric?
c) Is * anti-symmetric?
d) Is * transitive?
e) Is * an equivalence relation, a partial order, both, or neither?

Added by James H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Hoan Nguyen

04/17/2023

Step 1

We need to find a real number 0 < k ≠ 1 such that x1 = kx1 and x2 = kx2. If we choose k = 1, we have x1 = x1 and x2 = x2, which is true for all (x1, x2) in R^2. However, k must not be equal to 1, so * is not reflexive. b) To check if * is symmetric, we need to Show more…

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A certain relation * is defined on the set R^2 by (x1, x2) * (y1, y2) if and only if there exists a real number 0 < k <= 1 such that y2 = kx1 and x2 = ky1. For each of the following questions, be sure to provide a proof supporting your answer. a) Is * reflexive? b) Is * symmetric? c) Is * anti-symmetric? d) Is * transitive? e) Is * an equivalence relation, a partial order, both, or neither?