91Ó°ÊÓ

Which of these relations on \(\\{0,1,2,3\\}\) are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) \(\\{(0,0),(1,1),(2,2),(3,3)\\}\) b) \(\\{(0,0),(0,2),(2,0),(2,2),(2,3),(3,2),(3,3)\\}\) c) \(\\{(0,0),(1,1),(1,2),(2,1),(2,2),(3,3)\\}\) d) \(\\{(0,0),(1,1),(1,3),(2,2),(2,3),(3,1),(3,2),(3,3)\\}\) e) \(\\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0)\) \((2,2),(3,3) \\}\)

List the ordered pairs in the relation \(R\) from \(A=\\{0,1,2,3,4\\}\) to \(B=\\{0,1,2,3\\},\) where \((a, b) \in R\) if and only if $$ \begin{array}{ll}{\text { a) } a=b .} & {\text { b) } a+b=4} \\ {\text { c) } a>b .} & {\text { d) } a | b} \\ {\text { e) } \operatorname{gcd}(a, b)=1 .} & {\text { f) } \operatorname{lcm}(a, b)=2}\end{array} $$

Represent each of these relations on \(\\{1,2,3\\}\) with a matrix (with the elements of this set listed in increasing order). a) \(\\{(1,1),(1,2),(1,3)\\}\) b) \(\\{(1,2),(2,1),(2,2),(3,3)\\}\) c) \(\\{(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)\\}\) d) \(\\{(1,3),(3,1)\\}\)

Represent each of these relations on \(\\{1,2,3,4\\}\) with a matrix (with the elements of this set listed in increasing order). a) \(\\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\\}\) b) \(\\{(1,1),(1,4),(2,2),(3,3),(4,1)\\}\) c) \(\\{(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2)\) (3, \(4 ),(4,1),(4,2),(4,3) \\}\) d) \(\\{(2,4),(3,1),(3,2),(3,4)\\}\)

Let \(R\) be the relation \(\\{(a, b) | a \text { divides } b\\}\) on the set of integers. What is the symmetric closure of \(R ?\)

For each of these relations on the set \(\\{1,2,3,4\\},\) decide whether it is reflexive, whether it is symmetric, and whether it is antisymmetric, and whether it is transitive. $$ \begin{array}{l}{\text { a) }\\{(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)\\}} \\\ {\text { b) }\\{(1,1),(1,2),(2,1),(2,2),(3,3),(3,4)\\}} \\ {\text { c) }\\{(2,4),(4,2)\\}} \\ {\text { d) }\\{(1,2),(2,3),(3,4)\\}} \\ {\text { e) }\\{(1,1),(2,2),(3,3),(4,4)\\}} \\ {\text { f) }\\{(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)\\}}\end{array} $$

List the ordered pairs in the relations on \(\\{1,2,3,4\\}\) corresponding to these matrices (where the rows and columns correspond to the integers listed in increasing order). a) \(\left[\begin{array}{llll}{1} & {1} & {0} & {1} \\ {1} & {0} & {1} & {0} \\\ {0} & {1} & {1} & {1} \\ {1} & {0} & {1} & {1}\end{array}\right]\) b) \(\left[\begin{array}{llll}{1} & {1} & {1} & {0} \\ {0} & {1} & {0} & {0} \\\ {0} & {0} & {1} & {1} \\ {1} & {0} & {0} & {1}\end{array}\right]\) c) \(\left[\begin{array}{llll}{0} & {1} & {0} & {1} \\ {1} & {0} & {1} & {0} \\\ {0} & {1} & {0} & {1} \\ {1} & {0} & {1} & {0}\end{array}\right]\)

How can the directed graph representing the reflexive closure of a relation on a finite set be constructed from the directed graph of the relation?

Is \((S, R)\) a poset if \(S\) is the set of all people in the world and \((a, b) \in R,\) where \(a\) and \(b\) are people, if a) \(a\) is no shorter than \(b ?\) b) \(a\) weighs more than \(b\) ? c) \(a=b\) or \(a\) is a descendant of \(b ?\) d) \(a\) and \(b\) do not have a common friend?

Determine whether the relation \(R\) on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive, where \((a, b) \in R\) if and only if a) \(a\) is taller than \(b\). b) \(a\) and \(b\) were born on the same day. c) \(a\) has the same first name as \(b\) . d) \(a\) and \(b\) have a common grandparent.