91Ó°ÊÓ

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. $$A \odot(B \odot C)$$

Determine if the given elements are comparable in the poset \((A, |),\) where \(A=\\{1,2,3,6,9,18\\}\) and \(|\) denotes the divisibility relation. $$2,9$$

Using the following adjacency matrices of relations \(R\) and \(S\) on \(\\{a, b, c\\},\) find the adjacency matrices. $$M_{R}=\left[\begin{array}{lll}1 & 0 & 1 \\\0 & 1 & 0 \\\0 & 1 & 1\end{array}\right] \quad M_{S}=\left[\begin{array}{lll}0 & 1 & 1 \\\0 & 0 & 0 \\\1 & 0 & 1\end{array}\right]$$ $$\left(M_{R}\right)^{| 4\rfloor}$$

A FORTRAN program contains 10 variables, A through, and the following equivalence statement: EQUIVALENCE (A,B,C),(D,E),(F, B),(C,H). Find each class. \([B]\)

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. $$(A \odot B) \odot C$$

Determine if the given elements are comparable in the poset \((A, |),\) where \(A=\\{1,2,3,6,9,18\\}\) and \(|\) denotes the divisibility relation. $$3,18$$

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. $$(A \odot B) \odot C$$

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. $$\left[\begin{array}{llll}{0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0} \\\ {1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {1}\end{array}\right]$$

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

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 ). $$\\{a, b\\},\\{b, c\\}$$