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Chapter 1: Problem 6
Prove that \(n^{3}+5 n\) is divisible by 6 for all \(n \in \mathbb{N}\).
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Let \(g(x):=x^{2}\) and \(f(x):=x+2\) for \(x \in \mathbb{R}\), and let \(h\) be the composite function \(h:=g \circ f\). (a) Find the direct image \(h(E)\) of \(E:=\\{x \in \mathbb{R}: 0 \leq x \leq 1\\}\). (b) Find the inverse image \(h^{-1}(G)\) of \(G:=(x \in \mathbb{R}: 0 \leq x \leq 4\\}\).
(a) Suppose that \(f\) is an injection. Show that \(f^{-1}\) o \(f(x)=x\) for all \(x \in D(f)\) and that \(f \circ f^{-1}(y)=y\) for all \(y \in R(f)\) (b) If \(f\) is a bijection of \(A\) onto \(B\), show that \(f^{-1}\) is a bijection of \(B\) onto \(A\).
Find the largest nacural number \(m\) such that \(n^{3}-n\) is divisible by \(m\) for all \(n \in \mathbb{N}\). Prove your assertion.
(a) Show that if \(f: A \rightarrow B\) is injective and \(E \subseteq A\), then \(f^{-1}(f(E))=E\). Give an example to show that equality need not hold if \(f\) is not injective. (b) Show that if \(f: A \rightarrow B\) is surjective and \(H \subseteq B\), then \(f\left(f^{-1}(H)\right)=H .\) Give an example to show that equality need not hold if \(f\) is not surjective.
Prove that \(n<2^{n}\) for all \(n \in \mathbb{N}\)
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