91Ó°ÊÓ

Let \(A=\\{1,2,3,4\\},\) and let \(r\) be the relation \(\leq\) on \(A\). Draw a digraph for \(r\)

Determine which of the following are equivalence relations and/or partial ordering relations for the given sets: (a) \(A=\\{\) lines in the plane \(\\}\), and \(r\) defined by \(x r y\) if and only if \(x\) is parallel to \(y\). Assume every line is parallel to itself. (b) \(A=\mathbb{R}\) and \(r\) defined by \(x r y\) if and only if \(|x-y| \leq 7\).

What common relations on \(\mathbb{Z}\) are the transitive closures of the following relations? (a) \(a S b\) if and only if \(a+1=b\). (b) \(a R b\) if and only if \(|a-b|=2\).