91Ó°ÊÓ

A function \(f: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\) is defined as \(f(m, n)=(m+n, 2 m+n)\). Verify whether this function is injective and whether it is surjective.

Consider the set \(f=\\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: x+3 y=4\\} .\) Is this a function from \(\mathbb{Z}\) to \(\mathbb{Z} ?\) Explain.

Prove that the function \(f: \mathbb{R}-\\{2\\} \rightarrow \mathbb{R}-\\{5\\}\) defined by \(f(x)=\frac{5 x+1}{x-2}\) is bijective.

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 \(f \circ g\).

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.

Given a function \(f: A \rightarrow B\) and subsets \(W, X \subseteq A,\) prove \(f(W \cup X)=f(W) \cup f(X)\).

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.

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.

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 \(f \circ f\).

Prove the function \(f: \mathbb{R}-\\{1\\} \rightarrow \mathbb{R}-\\{1\\}\) defined by \(f(x)=\left(\frac{x+1}{x-1}\right)^{3}\) is bijective.