91Ó°ÊÓ

A number of binary relations are defined on the set \(A=\\{0,1,2,3\\}\). For each relation: a. Draw the directed graph. b. Determine whether the relation is reflexive. c. Determine whether the relation is symmetric. d. Determine whether the relation is transitive. Give a counterexample in each case in which the relation does not satisfy one of the properties. $$R_{1}=\\{(0,0),(0,1),(0,3),(1,1),(1,0),(2,3),(3,3)\\}$$

Each of the following is a relation on \(\\{0,1,2,3\\}\). Draw directed graphs for each relation, and indicate which relations are antisymmetric. a. \(R_{1}=\\{(0,0),(0,2),(1,0),(1,3),(2,2),(3,0),(3,1)\\}\) b. \(R_{2}=\\{(0,1),(0,2),(1,1),(1,2),(1,3),(2,2),(3,2)\\}\) c. \(R_{3}=\\{(0,0),(0,3),(1,0),(1,3),(2,2),(3,3),(3,2)\\}\) d. \(R_{4}=\\{(0,0),(1,0),(1,2),(1,3),(2,0),(2,1),(3,2)\), \((3,0)\\}\)

a. Use the Caesar cipher to encrypt the message WHERE SHALL WE MEET. b. Use the Caesar cipher to decrypt the message LQ WKH FDIHWHULD.

Each of the following partitions of \(\\{0,1,2,3,4\\}\) induces a relation \(R\) on \(\\{0,1,2,3,4\\}\). In each case, find the ordered pairs in \(R\). a. \(\\{0,2\\},\\{1\\},\\{3,4\\}\) b. \(\\{0\\},(1,3,4\\},\\{2\\}\) c. \(\\{0\\},\\{1,2,3,4\\}\) in 2-12, the relation \(R\) is an equivalence relation on the set \(A\). Find the distinct equivalence classes of \(R\).

As in Example 10.1.2, the congruence modulo 2 relation \(E\) is defined from \(\mathbf{Z}\) to \(\mathbf{Z}\) as follows: For all integers \(m\) and \(n, m E n \Leftrightarrow m-n\) is even, a. Is \(0 E 0\) ? Is \(5 E\) 2? Is \((6,6) \in E\) ? Is \((-1,7) \in E\) ? b. Prove that for any even integer \(n, n E 0\).

Let \(R\) be the "less than" relation on the set \(\mathbf{R}\) of all real numbers: For all \(x, y \in \mathbf{R}\), $$ x R y \quad \Leftrightarrow \quad x

Prove that for all integers \(m\) and \(n, m-n\) is even if, and only if, both \(m\) and \(n\) are even or both \(m\) and \(n\) are odd.

A number of binary relations are defined on the set \(A=\\{0,1,2,3\\}\). For each relation: a. Draw the directed graph. b. Determine whether the relation is reflexive. c. Determine whether the relation is symmetric. d. Determine whether the relation is transitive. Give a counterexample in each case in which the relation does not satisfy one of the properties. $$R_{4}=\\{(1,2),(2,1),(1,3),(3,1)\\}$$

Prove that the distinct equivalence classes of the relation of congruence modulo \(n\) are the sets \([0],[1],[2], \ldots,[n-1]\), where for each \(a=0,1,2, \ldots, n-1\), $$ [a]=\\{m \in Z \mid m \equiv a(\bmod n)\\} $$

Define a relation \(R\) on the set \(\mathbf{Z}\) of all integers as follows: For all \(m, n \in \mathbf{Z}\), \(\begin{aligned} m R n & \Leftrightarrow & \text { every prime factor of } m \\\ & \text { is a prime factor of } n . \end{aligned}\) Is \(R\) a partial order relation? Prove or give a counterexample.