91Ó°ÊÓ

Using the boolean matrices $$ A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right], B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right], \text { and } C=\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right] $$ find each. $$B \odot C \odot A$$

Define a relation \(R\) on the set of U.S. cities as follows: \(x R y\) if a direct communication link exists from city \(x\) to city \(y .\) How would you interpret \(R^{2} ? R^{n} ?\)

The adjacency matrix of a relation \(R\) on \(\\{a, b, c, d\\}\) is given. In each case, compute the boolean matrices \(W_{1}\) and \(W_{2}\) in Warshall's algorithm. $$\left[\begin{array}{llll} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right]$$

Using the boolean matrices $$A=\left[\begin{array}{lll}{1} & {0} & {1} \\ {0} & {0} & {0} \\ {1} & {1} & {0}\end{array}\right], B=\left[\begin{array}{lll}{0} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {0}\end{array}\right], \text { and } C=\left[\begin{array}{lll}{0} & {0} & {0} \\ {0} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right]$$ find each. $$B \odot C \odot A$$

In Exercises \(16-18,\) the adjacency matrix of a relation \(R\) on \(\\{a, b, c, d\\}\) is given. In each case, compute the boolean matrices \(W_{1}\) and \(W_{2}\) in Warshall's algorithm. $$\left[\begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {1} \\\ {0} & {1} & {0} & {1} \\ {0} & {1} & {0} & {1}\end{array}\right]$$

Using the boolean matrices $$A=\left[\begin{array}{lll}{1} & {0} & {1} \\ {0} & {0} & {0} \\ {1} & {1} & {0}\end{array}\right], B=\left[\begin{array}{lll}{0} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {0}\end{array}\right], \text { and } C=\left[\begin{array}{lll}{0} & {0} & {0} \\ {0} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right]$$ find each. $$A \odot A \odot A$$

Determine if the given elements are comparable in the poset \((A, \subseteq),\) where \(A\) denotes the power set of \(\\{a, b, c\\}\) (see Example 7.58 ). $$\\{a, b\\},\\{b\\}$$

In Exercises \(17-19,\) the adjacency matrices of three relations on \(\\{a, b, c\\}\) are given. Determine if each relation is reflexive, symmetric, or antisymmetric. $$ \left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right] $$

Using the boolean matrices $$ A=\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right], B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right], \text { and } C=\left[\begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right] $$ find each. $$A \odot A \odot A$$

The adjacency matrix of a relation \(R\) on \(\\{a, b, c, d\\}\) is given. In each case, compute the boolean matrices \(W_{1}\) and \(W_{2}\) in Warshall's algorithm. $$\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 \end{array}\right]$$