Assignment 13 (pdf)

Due in class Thursday of Week 4

1. Consider the set \(A=\{1,2,3,4,5\}\) consisting of five elements. We define a relation \(R\) on \(A\) via \(x R y\) if and only if the product \(xy\) is even. Determine if \(R\) is an equivalence relation, and if so find the equivalence classes.
2. Suppose \(U\) is a set of five elements, \(U=\{a,b,c,d,e\}\). We then consider the set \(A = P(U)\), the powerset of \(U\), which consists of all subsets of \(U\). We define on \(A\) two relations (so these relations take as elements subsets of \(U\)). The first, \(R_1\), is defined by saying that two subsets \(B_1\), \(B_2\) of \(U\) are related if and only if their intersection \(B_1\cap B_2\) is nonempty. The second, \(R_2\), is defined by saying that two subsets \(B_1\), \(B_2\) of \(U\) are related if and only if their union \(B_1\cup B_2\) is nonempty. Is \(R_1\) an equivalence relation? Is \(R_2\) an equivalence relation? Prove or disprove.
3. Consider a set \(A\) and a relation \(R\) on \(A\) with the following properties:
   • \(R\) is symmetric and transitive.
   • For every \(x\) in \(A\) there is a \(y\) in \(A\) such that \(xRy\) (i.e. every element is related to some element, which may or may not be equal to it).
   Then show that \(R\) is also reflexive.