91Ó°ÊÓ

Let \(R\) be the relation on the set of all students containing the ordered pair \((a, b)\) if \(a\) and \(b\) are in at least one common class and \(a \neq b .\) When is \((a, b)\) in $$ \begin{array}{llll}{\text { a) } R^{2} ?} & {\text { b) } R^{3} ?} & {\text { c) } R^{*} ?}\end{array} $$

Draw the Hasse diagram for the less than or equal to relation on \(\\{0,2,5,10,11,15\\} .\)

Draw the Hasse diagram for divisibility on the set a) \(\\{1,2,3,4,5,6\\}\) b) \(\\{3,5,7,11,13,16,17\\}\) c) \(\\{2,3,5,10,11,15,25\\}\) d) \(\\{1,3,9,27,81,243\\}\)

Determine whether the relations represented by these zero-one matrices are equivalence relations. $$ \text { a) }\left[\begin{array}{lll}{1} & {1} & {1} \\ {0} & {1} & {1} \\\ {1} & {1} & {1}\end{array}\right] \quad \text { b) }\left[\begin{array}{llll}{1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {1} \\\ {1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {1}\end{array}\right] \text { c) }\left[\begin{array}{llll}{1} & {1} & {1} & {0} \\ {1} & {1} & {1} & {0} \\\ {1} & {1} & {1} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right] $$

Draw the Hasse diagram for inclusion on the set \(P(S),\) where \(S=\\{a, b, c, d\\}\)

How many different relations are there from a set with \(m\) elements to a set with \(n\) elements?

Let \(R\) be a relation from a set \(A\) to a set \(B\) . The inverse relation from \(B\) to \(A,\) denoted by \(R^{-1}\) , is the set of ordered pairs \(\\{(b, a) |(a, b) \in R\\} .\) The complementary relation \(\overline{R}\) is the set of ordered pairs \(\\{(a, b) |(a, b) \notin R\\}\). Let \(R\) be the relation \(R=\\{(a, b) | a \text { divides } b\\}\) on the set of positive integers. Find \(\begin{array}{ll}{\text { a) } R^{-1} .} & {\text { b) } \overline{R}}\end{array}\)

What is the covering relation of the partial ordering \(\\{(a, b) | a \text { divides } b\\}\) on \(\\{1,2,3,4,6,12\\} ?\)

Let \(R\) be a relation from a set \(A\) to a set \(B\) . The inverse relation from \(B\) to \(A,\) denoted by \(R^{-1}\) , is the set of ordered pairs \(\\{(b, a) |(a, b) \in R\\} .\) The complementary relation \(\overline{R}\) is the set of ordered pairs \(\\{(a, b) |(a, b) \notin R\\}\). Let \(R\) be the relation on the set of all states in the United States consisting of pairs \((a, b)\) where state \(a\) borders state \(b .\) Find \(\begin{array}{ll}{\text { a) } R^{-1}} & {\text { b) } \overline{R}}\end{array}\)

Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is a) reflexive and transitive. b) symmetric and transitive. c) reflexive, symmetric, and transitive.