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. $$A \odot A \odot A$$

Arrange the following pairs from the poset \(\mathbb{N} \times \mathbb{N}\) in lexicographic order. $$(3,5),(2,3)$$

Using the boolean matrix $$ A=\left[\begin{array}{lll} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ find \(A^{[3]}\) and \(A^{|5|}\)

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

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

Using the boolean matrix $$A=\left[\begin{array}{lll}{1} & {0} & {1} \\ {1} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]$$ \(A^{|3|} \text { and } A^{|5|}\)

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

Arrange the following pairs from the poset \(N \times \mathbb{N}\) in lexicographic order. \((3,5),(2,3)\)

The complement and inverse of a relation \(R\) from a set \(A\) to a set \(B,\) denoted by \(R^{\prime}\) and \(R^{-1}\) respectively, are defined as follows: \(R^{\prime}=\) \(\\{(a, b) | a R b\\}\) and \(R^{-1}=\\{(a, b) | b R a\\} .\) So \(R^{\prime}\) consists of all elements in \(A \times B\) that are not in \(R,\) whereas \(R^{-1}\) consists of all elements \((a, b)\) where \((b, a) \in R .\) Using the relations \(R=\\{(a, a),(a, b),(b, c),(c, c)\\}\) and \(S=\\{(a, a),(b, b),(b, c),(c, a)\\}\) on \(\\{a, b, c\\},\) find each. $$R^{-1}$$

Represent each relation \(R\) on the given set \(A\) in a digraph. $$\\{(a, b) | a < b\\},\\{2,3,5\\}$$