Homework 8: Equivalence Relations and Quotient Sets

1. Let \(X\) be a nonempty set. A relation \(R\) on \(X\) is said to be circular if, for all \(x,y,z\in X\), \(x\,R\,y\) and \(y\,R\,z\) implies \(z\,R\, x\). Prove that a relation \(R\) on \(X\) is an equivalence relation if and only if \(R\) is reflexive and circular.
2. Let \(X\) be the set of directed line segments \(PQ\) (with initial point \(P\) and terminal point \(Q\)) in \(\bR^2\). Give an equivalence relation \(\sim\) for which \(X/\sim\) is the usual set of vectors in \(\bR^2\).
3. Consider \(\bR\) with relation \(Q = \{(x,y)\in \bR\times \bR: x-y\in \bZ\}\).
   1. Verify that \(Q\) is an equivalence relation.
   2. Find the equivalence classes.
   3. Identify \(\bR/Q\) with an interval in \(\bR\).
   4. Identify \(\bR/Q\) with \(S^1\).
4. Consider \(\bR^2\) with relation \((a,b)\sim (c,d)\) if \(a+2d=c+2b\).
   1. Verify that \(\sim\) is an equivalence relation.
   2. What are the equivalence classes?
5. Let \(X=\{(a,b)\in \bZ\times \bZ\mid b\neq 0\}\). Define relation \(\sim\) on \(X\) by \((a,b)\sim (c,d)\) if \(ad=bc\).
   1. Verify that \(\sim\) is an equivalence relation.
   2. Give three elements in \([(1,2)]\).
   3. \(X/\sim\) is sometimes used to define a set with which you are familiar. Give the set.
6. (Number Theory) Consider \(\bZ\) with relation \(a\sim b\) if \(5\mid a-b\).
   1. Verify that \(\sim\) is an equivalence relation.
   2. Find the equivalence classes.
   3. What is the cardinality of \(\bZ/\sim\)?
7. (Linear Algebra) Let \(V\) be a vector space and \(W\) a subspace. Consider the relation \(\sim\) on \(V\) defined by \(\bu \sim \bv\) if \(\bu - \bv\in W\).
   1. Verify that \(\sim\) is an equivalence relation. The corresponding quotient \(V/\sim\) is usually denoted by \(V/W\).
   2. Let \(V=\bR^2\) and \(W=\) Span\(\vtwo{1}{1}\), a subspace of \(V\). What are the equivalence classes? What is \(V/W\)?