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Magazine Problem 6
Given a function \(f: A \rightarrow B\) and a subset \(Y \subseteq B,\) is \(f\left(f^{-1}(Y)\right)=Y\) always true? Prove or give a counterexample.
Problem 6
Consider the functions \(f, g: \mathbb{R} \rightarrow \mathbb{R}\) defined as \(f(x)=\frac{1}{x^{2}+1}\) and \(g(x)=3 x+2 .\) Find the formulas for \(g \circ f\) and \(f \circ g\).
Problem 7
Consider the set \(f=\\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: 3 x+y=4\\} .\) Is this a function from \(\mathbb{Z}\) to \(\mathbb{Z} ?\) Explain.
Problem 7
A function \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}\) is defined as \(f(m, n)=2 n-4 m .\) Verify whether this function is injective and whether it is surjective.
Problem 7
Consider the functions \(f, g: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) defined as \(f(m, n)=\left(m n, m^{2}\right)\) and \(g(m, n)=(m+1, m+n)\). Find the formulas for \(g \circ f\) and \(f \circ g\).
Problem 7
Given a function \(f: A \rightarrow B\) and subsets \(W, X \subseteq A,\) prove \(f(W \cap X) \subseteq f(W) \cap f(X)\).
Problem 7
Prove or disprove: Any subset \(X \subseteq\\{1,2,3, \ldots, 2 n\\}\) with \(|X|>n\) contains two (unequal) elements \(a, b \in X\) for which \(a \mid b\) or \(b \mid a\).
Problem 8
Consider the functions \(f, g: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) defined as \(f(m, n)=(3 m-4 n, 2 m+n)\) and \(g(m, n)=(5 m+n, m) .\) Find the formulas for \(g \circ f\) and \(f \circ g\).
Problem 8
Given a function \(f: A \rightarrow B\) and subsets \(W, X \subseteq A,\) then \(f(W \cap X)=f(W) \cap f(X)\) is false in general. Produce a counterexample.
Problem 8
Is the function \(\theta: \mathscr{P}(\mathbb{Z}) \rightarrow \mathscr{P}(\mathbb{Z})\) defined as \(\theta(X)=\bar{X}\) bijective? If so, find \(\theta^{-1}\).
