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Chapter 3: Problem 26
\((x, y) \in R\) if \(x y=2\)
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For the sequence a defined by \(a_{n}=\frac{n-1}{n^{2}(n-2)^{2}}, \quad n \geq 3\) and the sequence \(z\) defined by \(z_{n}=\sum_{i=3}^{n} a_{i}\). Is \(z\) increasing?
Sometimes we generalize the notion of sequence as defined in this section by allowing more general indexing. Suppose that \(\left\\{a_{i j}\right\\}\) is a sequence indexed over pairs of positive integers. Prove that $$ \sum_{i=1}^{n}\left(\sum_{j=i}^{n} a_{i j}\right)=\sum_{j=1}^{n}\left(\sum_{i=1}^{j} a_{i j}\right) $$
Use the following definitions. Let \(U\) be a universal set and let \(X \subseteq U\). Define $$ C_{X}(x)=\left\\{\begin{array}{ll} 1 & \text { if } x \in X \\ 0 & \text { if } x \notin X . \end{array}\right. $$ We call \(C_{X}\) the characteristic function of \(X(\) in \(U) .\) (A look ahead at the next Problem-Solving Corner may help in understanding the following exercises.) Prove that if \(X \subseteq Y,\) then \(C_{X}(x) \leq C_{Y}(x)\) for all \(x \in U\)
For the sequence \(r\) defined by $$r_{n}=3 \cdot 2^{n}-4 \cdot 5^{n}, \quad n \geq 0$$. Find a formula for \(r_{n-1}\).
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 \(|a-b|<\varepsilon\).
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