91Ó°ÊÓ

For the sequence b defined by \(b_{n}=n(-1)^{n}, n \geq 1\). Find a formula for the sequence \(c\) defined by $$ c_{n}=\sum_{i=1}^{n} b_{i} $$

Given $$ f=\left\\{\left(x, x^{2}\right) \mid x \in X\right\\} $$ a function from \(X=\\{-5,-4, \ldots, 4,5\\}\) to the set of integers, write \(f\) as a set of ordered pairs and draw the arrow diagram of \(f\). Is \(f\) one-to-one or onto?

For the sequence b defined by \(b_{n}=n(-1)^{n}, n \geq 1\). Find a formula for the sequence \(d\) defined by $$ d_{n}=\prod_{i=1}^{n} b_{i} $$

If the statement is true for all relations \(R_{1}\) and \(R_{2}\) on an arbitrary set \(X,\) prove it; otherwise, give \(a\) counterexample. $$ \rho\left(R_{1} \cup R_{2}\right)=\rho\left(R_{1}\right) \cup \rho\left(R_{2}\right) $$

Determine whether each relation \(R\) defined on the collection of all nonempty subsets of real numbers is reflexive, symmetric, antisymmetric, transitive, and/or a partial order. \((A, B) \in R\) if for every \(a \in A\) and \(\varepsilon>0,\) there exists \(b \in B\) with \(|a-b|<\varepsilon\).

For the sequence b defined by \(b_{n}=n(-1)^{n}, n \geq 1\). Is \(b\) increasing?

How many functions are there from \\{1,2\\} to \(\\{a, b\\}\) ? Which are one- to-one? Which are onto?

Given $$ f=\\{(a, b),(b, a),(c, b)\\} $$ a function from \(X=\\{a, b, c\\}\) to \(X\) : (a) Write \(f \circ f\) and \(f \circ f \circ f\) as sets of ordered pairs. (b) Define $$ f^{n}=f \circ f \circ \cdots \circ f $$ to be the \(n\) -fold composition of \(f\) with itself. Write \(f^{9}\) and \(f^{623}\) as sets of ordered pairs.

For the sequence b defined by \(b_{n}=n(-1)^{n}, n \geq 1\). Is \(b\) decreasing?

Let \(f\) be the function from \(X=\\{0,1,2,3,4\\}\) to \(X\) defined by $$ f(x)=4 x \bmod 5 $$ Write \(f\) as a set of ordered pairs and draw the arrow diagram of \(f\). Is \(f\) one-to-one? Is \(f\) onto?