91Ó°ÊÓ

Write the relation as a set of ordered pairs. $$\begin{array}{rll}\hline 8840 & \text { Hammer } \\\9921 & \text { Pliers } \\\452 & \text { Paint } \\\2207 & \text { Carpet } \\\\\hline\end{array}$$

Find the matrix of the relation \(R\) on \(X\) relative to the ordering given. \(R=\\{(1,2),(2,3),(3,4),(4,5)\\} ;\) ordering of \(X: 1,2,3,4,5\)

Draw the digraph of the relation. The relation \(R=\\{(1,2),(2,3),(3,4),(4,1)\\}\) on \\{1,2,3,4\\}

Determine whether each set in is a function from \(X=\\{1,2,3,4\\}\) to \(Y=\\{a, b, c, d\\} .\) If it is a function, find its domain and range, draw its arrow diagram, and determine if it is one-to-one, onto, or both. If it is both one- to-one and onto, give the description of the inverse function as a set of ordered pairs, draw its arrow diagram, and give the domain and range of the inverse function. $$ \\{(1, b),(2, b),(3, b),(4, b)\\} $$

List the members of the equivalence relation on \(\\{1,2,3,4\\}\) defined by the given partition. Also, find the equivalence classes \([1], [2], [3]\), and \([4]\). $$ \\{\\{1,2\\},\\{3,4\\}\\} $$

Determine whether each function in is one-to-one, onto, or both. Prove your answers. The domain of each function is the set of all integers. The codomain of each function is also the set of all integers. $$ f(n)=n+1 $$

Determine whether each function in is one-to-one, onto, or both. Prove your answers. The domain of each function is the set of all integers. The codomain of each function is also the set of all integers. $$ f(n)=n^{2}-1 $$

For the sequence t defined by \(t_{n}=2 n-1, \quad n \geq 1\). Find a formula that represents this sequence as a sequence whose lower index is 0.

Determine whether each function in is one-to-one, onto, or both. Prove your answers. The domain of each function is the set of all integers. The codomain of each function is also the set of all integers. $$ f(n)=2 n $$

Determine whether each relation defined on the set of positive integers is reflexive, symmetric, antisymmetric. transitive, and/or a partial order. \((x, y) \in R\) if \(x y=1\)