91Ó°ÊÓ

Which of these are posets? a) \((\mathbf{Z},=)\) b) \((\mathbf{Z}, \neq)\) c) \((\mathbf{Z}, \geq)\) d) \((\mathbf{Z}, \not{|})\)

Which of these are posets? a) \((\mathbf{R},=)\) b) \((\mathbf{R},<)\) c) \((\mathbf{R}, \leq)\) d) \((\mathbf{R}, \neq)\)

Determine whether the relation \(R\) on the set of all real numbers is reflexive, symmetric, antisymmetric, and/or transitive, where \((x, y) \in R\) if and only if a) \(x+y=0\) b) \(x=\pm y\) c) \(x-y\) is a rational number d) \(x=2 y\) e) \(x y \geq 0\) f) \(x y=0\) g) \(x=1\) h) \(x=1\) or \(y=1\)

Show that the relation of logical equivalence on the set of all compound propositions is an equivalence relation. What are the equivalence classes of \(\mathbf{F}\) and of \(\mathbf{T} ?\)

Determine whether the relation \(R\) on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where \((x, y) \in R\) if and only if a) \(x \neq y\) b) \(x y \geq 1\) c) \(x=y+1\) or \(x=y-1\) d) \(x \equiv y(\bmod 7)\) e) \(x\) is a multiple of \(y\) f) \(x\) and \(y\) are both negative or both nonnegative. g) \(x=y^{2}\) h) \(x \geq y^{2}\)

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

Determine whether the relations represented by these zero–one matrices are partial orders. a) \(\left[\begin{array}{lll}{1} & {0} & {1} \\ {1} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]\) b) \(\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {1} & {0} & {1}\end{array}\right]\) c) \(\left[\begin{array}{cccc}{1} & {0} & {1} & {0} \\ {0} & {1} & {1} & {0} \\\ {0} & {0} & {1} & {1} \\ {1} & {1} & {0} & {1}\end{array}\right]\)

Show that the relation \(R=\emptyset\) on the empty set \(S=\emptyset\) is reflexive, symmetric, and transitive.

How many nonzero entries does the matrix representing the relation \(R\) on \(A=\\{1,2,3, \ldots, 1000\\}\) consisting of the first 1000 positive integers have if \(R\) is a) \(\\{(a, b) | a \leq b\\} ?\) b) \(\\{(a, b) | a=b \pm 1\\} ?\) c) \(\\{(a, b) | a+b=1000\\} ?\) d) \(\\{(a, b) | a+b \leq 1001\\} ?\) e) \(\\{(a, b) | a \neq 0\\} ?\)

Give an example of a relation on a set that is a) both symmetric and antisymmetric. b) neither symmetric nor antisymmetric.