18.7: Exercises
- Page ID
- 93871
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(y_1 - x_1^2 = y_2 - x_2^2\text{.}\)
- Verify that \(\mathord{\equiv}\) is an equivalence relation.
- 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.
- Prove that \(R\) is an equivalence relation on \(V\text{.}\)
- 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.
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Does the expression \(x^2 + y^2\) remind you of anything from geometry?