18.7: Exercises

Exercise $\PageIndex{1}$

Let $\mathord{\equiv}$ represent the relation on $\mathbb{R}\times\mathbb{R}$ where $(x_1,y_1) \equiv (x_2,y_2)$ means $y_1 - x_1^2 = y_2 - x_2^2\text{.}$

1. Verify that $\mathord{\equiv}$ is an equivalence relation.
2. Describe the equivalence classes $[(0,0)]\text{,}$ $[(0,1)]\text{,}$ and $[(1,0)]$ geometrically as sets of points in the plane.

Exercise $\PageIndex{2}$

Given a connected (undirected) graph $G\text{,}$ we can define a relation on the set $V$ of vertices in $G$ as follows: let $v_1 R v_2$ mean that there exists a trail within $G$ beginning at vertex $v_1$ and ending at vertex $v_2$ that traverses an even number of edges.

1. Prove that $R$ is an equivalence relation on $V\text{.}$
2. Determine the equivalence classes for this relation when $G$ is the graph below.

Figure $\PageIndex{1}$

Equivalence relations and classes.

In each of Exercises 3–12, you are given a set $A$ and a relation $R$ on $A\text{.}$ Determine whether $R$ is an equivalence relation, and, if it is, describe its equivalence classes. Try to be more descriptive than just “$[a]$ is the set of all elements that are equivalent to $a\text{.}$”

 

Exercise $\PageIndex{3}$

$A = \{a, b, c\} \text{;}$ $R = \{(a,a),(b,b),(c,c),(a,b),(b,a)\} \text{.}$

Exercise $\PageIndex{4}$

$A = \{-1, 0, 1\} \text{;}$ $R = \{(x,y) | x^2 = y^2\} \text{.}$

Exercise $\PageIndex{5}$

$A$ is the power set of some set; $R$ is the subset relation.

Exercise $\PageIndex{6}$

$A = \mathbb{R} \text{;}$ $x_1 \mathrel{R} x_2$ means $f(x_1) = f(x_2) \text{,}$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ is the function $f(x) = x^2 \text{.}$

Exercise $\PageIndex{7}$

$A$ is some abstract set; $a_1 \mathrel{R} a_2$ means $f(a_1) = f(a_2)\text{,}$ where $f: A \rightarrow B$ is an arbitrary function with domain $A\text{.}$

Exercise $\PageIndex{8}$

$A$ is the set of all “formal” expressions $a/b\text{,}$ where $a,b$ are integers and $b$ is nonzero; $(a/b) \mathrel{R} (c/d)$ means $ad = bc \text{.}$

Note: Do not think of $a/b$ as a fraction in the usual way; instead think of it as a collection of symbols consisting of two integers in a specific order with a forward slash between them.

Exercise $\PageIndex{9}$

$A$ is the power set of some finite set; $X \mathrel{R} Y$ means $\vert X \vert = \vert Y \vert \text{.}$

Exercise $\PageIndex{10}$

$A$ is the set of all straight lines in the plane; $L_1 \mathrel{R} L_2$ means $L_1$ is parallel to $L_2\text{.}$

Exercise $\PageIndex{11}$

$A$ is the set of all straight lines in the plane; $L_1 \mathrel{R} L_2$ means $L_1$ is perpendicular to $L_2\text{.}$

Exercise $\PageIndex{12}$

$A = \mathbb{R} \times \mathbb{R} \text{;}$ $(x_1,y_1) \mathrel{R} (x_2,y_2)$ means $x_1^2 + y_1^2 = x_2^2 + y_2^2 \text{.}$

Hint.

Does the expression $x^2 + y^2$ remind you of anything from geometry?