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 9
Consider the function \(f: \mathbb{R} \times \mathbb{N} \rightarrow \mathbb{N} \times \mathbb{R}\) defined as \(f(x, y)=(y, 3 x y) .\) Check that this is bijective; find its inverse.
Problem 9
Consider the functions \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}\) defined as \(f(m, n)=m+n\) and \(g: \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) defined as \(g(m)=(m, m)\). Find the formulas for \(g \circ f\) and \(f \circ g\).
Problem 9
Given a function \(f: A \rightarrow B\) and subsets \(W, X \subseteq A,\) prove \(f(W \cup X)=f(W) \cup f(X)\).
Problem 9
Prove that the function \(f: \mathbb{R}-\\{2\\} \rightarrow \mathbb{R}-\\{5\\}\) defined by \(f(x)=\frac{5 x+1}{x-2}\) is bijective.
Problem 9
Consider the set \(f=\left\\{\left(x^{2}, x\right): x \in \mathbb{R}\right\\}\). Is this a function from \(\mathbb{R}\) to \(\mathbb{R}\) ? Explain.
Problem 10
Consider the set \(f=\left\\{\left(x^{3}, x\right): x \in \mathbb{R}\right\\} .\) Is this a function from \(\mathbb{R}\) to \(\mathbb{R} ?\) Explain.
Problem 10
Consider the function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by the formula \(f(x, y)=\left(x y, x^{3}\right) .\) Find a formula for \(f \circ f\).
Problem 10
Given \(f: A \rightarrow B\) and subsets \(Y, Z \subseteq B,\) prove \(f^{-1}(Y \cap Z)=f^{-1}(Y) \cap f^{-1}(Z)\).
