Let \(\mathord{\equiv}\) represent the relation on \(\R\times\R\) where \((x_1,y_1) \equiv (x_2,y_2)\) means \(y_1 - x_1^2 = y_2 - x_2^2\text{.}\)
(a)
Verify that \(\mathord{\equiv}\) is an equivalence relation.
(b)
Describe the equivalence classes \([(0,0)]\text{,}\)\([(0,1)]\text{,}\) and \([(1,0)]\) geometrically as sets of points in the plane.
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.
(a)
Prove that \(R\) is an equivalence relation on \(V\text{.}\)
(b)
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 “\(\eqclass{a}\) is the set of all elements that are equivalent to \(a\text{.}\)”
\(A\) is the power set of some set; \(R\) is the subset relation.
6.
\(A = \R \text{;}\)\(x_1 \mathrel{R} x_2 \) means \(f(x_1) = f(x_2) \text{,}\) where \(\funcdef{f}{\R}{\R}\) is the function \(f(x) = x^2 \text{.}\)
7.
\(A\) is some abstract set; \(a_1 \mathrel{R} a_2\) means \(f(a_1) = f(a_2)\text{,}\) where \(\funcdef{f}{A}{B}\) is an arbitrary function with domain \(A\text{.}\)
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.
9.
\(A\) is the power set of some finite set; \(X \mathrel{R} Y\) means \(\card{X} = \card{Y} \text{.}\)
10.
\(A\) is the set of all straight lines in the plane; \(L_1 \mathrel{R} L_2\) means \(L_1\) is parallel to \(L_2\text{.}\)
11.
\(A\) is the set of all straight lines in the plane; \(L_1 \mathrel{R} L_2\) means \(L_1\) is perpendicular to \(L_2\text{.}\)
