91Ó°ÊÓ

Given \(f: A \rightarrow B\) and subsets \(Y, Z \subseteq B,\) prove \(f^{-1}(Y \cap Z)=f^{-1}(Y) \cap f^{-1}(Z)\).

Given \(f: A \rightarrow B\) and subsets \(Y, Z \subseteq B,\) prove \(f^{-1}(Y \cup Z)=f^{-1}(Y) \cup f^{-1}(Z)\).

Consider the function \(\theta:\\{0,1\\} \times \mathbb{N} \rightarrow \mathbb{Z}\) defined as \(\theta(a, b)=(-1)^{a} b .\) Is \(\theta\) injective? Is it surjective? Bijective? Explain.

Is the set \(\theta=\left\\{(X,|X|): X \subseteq \mathbb{Z}_{5}\right\\}\) a function? If so, what is its domain and range?

Consider the function \(\theta:\\{0,1\\} \times \mathbb{N} \rightarrow \mathbb{Z}\) defined as \(\theta(a, b)=a-2 a b+b .\) Is \(\theta\) injective? Is it surjective? Bijective? Explain.

Consider \(f: A \rightarrow B\). Prove that \(f\) is injective if and only if \(X=f^{-1}(f(X))\) for all \(X \subseteq A .\) Prove that \(f\) is surjective if and only if \(f\left(f^{-1}(Y)\right)=Y\) for all \(Y \subseteq B\).

Is the set \(\theta=\\{((x, y),(3 y, 2 x, x+y)): x, y \in \mathbb{R}\\}\) a function? If so, what is its domain and range? What can be said about the codomain?

Consider the function \(f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) defined by the formula \(f(x, y)=\left(x y, x^{3}\right) .\) Is \(f\) injective? Is it surjective? Bijective? Explain.

Let \(f: A \rightarrow B\) be a function, and \(X \subseteq A .\) Prove or disprove: \(f\left(f^{-1}(f(X))\right)=f(X)\).

Let \(f: A \rightarrow B\) be a function, and \(Y \subseteq B\). Prove or disprove: \(f^{-1}\left(f\left(f^{-1}(Y)\right)\right)=f^{-1}(Y)\).