# 18.7: Exercises

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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.
###### 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?

This page titled 18.7: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.