18.6: Activities
- Page ID
- 93870
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Activity \(\PageIndex{1}\)
For each of the relations provided, carry out the following steps.
- Verify that the relation is an equivalence relation on the set \(A\text{.}\)
- Consider a few example equivalence classes, for the specific example representative elements provided (if applicable). What other elements are in that class?
- Devise a general way to describe every equivalence class, using your experience from the example classes already considered (if applicable). Make your class descriptions more meaningful than just “all elements equivalent to a specific representative element.”
- List/describe all elements in the quotient \(A/\equiv\text{.}\)
- Relation \(\mathord{\equiv}\) on \(A = \mathbb{Z}\text{,}\) where \(m \equiv n\) means \(m^2 = n^2\text{.}\) Example equivalence classes for \(1, 10, -2, 0\text{.}\)
- Relation \(\mathord{\equiv}\) on \(A = \mathbb{R} \times \mathbb{R}\text{,}\) where \((x_1,y_1) \equiv (x_2,y_2)\) means \(x_1^2 + y_1^2 = x_2^2 + y_2^2\text{.}\) Example equivalence classes for \((1,1), (3,4), (\sqrt{2}/2,-\sqrt{2}/2), (0,0)\text{.}\)
- Relation \(\mathord{\equiv}\) on \(A = \mathbb{R} \times \mathbb{R}\text{,}\) where \((x_1,y_1) \equiv (x_2,y_2)\) means \(y_1^2 - x_1 = y_2^2 - x_2\text{.}\) Example equivalence classes for \((0,0), (0,1), (1,-1)\text{.}\)
- Relation \(\mathord{\equiv}\) on \(A = \mathscr{P} (\{a,b,c,d\})\text{,}\) where \(X \equiv Y\) means \(\vert X^C \vert = \vert Y^C \vert\text{.}\) Example equivalence classes for \(\emptyset, \{a\}, \{a,b\}, \{a,b,c\}, \{a,b,c,d\}\text{.}\)
- Relation \(\mathord{\equiv}\) on the vertex set \(A = V\) of a graph \(G\text{,}\) where \(v \equiv v'\) means there exists a path in \(G\) from \(v\) to \(v'\text{.}\)
- Given function \(f: A \rightarrow B\text{,}\) the relation \(\mathord{\equiv}\) on the domain \(A\text{,}\) where \(a_1 \equiv a_2\) means \(f(a_1) = f(a_2)\text{.}\)
Activity \(\PageIndex{1}\)
A sequence from a set \(A\) could also be called an ordered list. For example, given distinct \(a_1,a_2 \in A\text{,}\) the finite sequences \(a_1,a_1,a_2\) and \(a_1,a_2,a_1\) are different sequences, because order matters in a sequence. However, as an unordered list, \(a_1,a_1,a_2\) is the same as \(a_1,a_2,a_1\text{.}\)
Write \(\mathscr{S}_A\) for the set of all finite sequences from \(A\text{.}\) Devise an equivalence relation \(\mathord{\equiv}\) on \(\mathscr{S}_A\) such that the quotient set \(\mathscr{S}_A / \mathord{\equiv}\) represents the set of all finite unordered lists from \(A\text{.}\)
- Hint.
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When should two different finite sequences be considered equivalent as unordered lists?
Activity \(\PageIndex{1}\)
Suppose \(\mathord{\equiv}\) and \(\mathord{\equiv}'\) are equivalence relations on a set \(A\text{.}\) Determine which of the following are also equivalence relations.
- \(\mathord{\equiv}^C\)
- \(\mathord{\equiv} \cup \mathord{\equiv}'\)
- \(\mathord{\equiv} \cap \mathord{\equiv}' \)
See Activity 17.5.4.