91Ó°ÊÓ

Consider the logarithm function \(\ln :(0, \infty) \rightarrow \mathbb{R}\). Decide whether this function is injective and whether it is surjective.

Consider the cosine function \(\cos : \mathbb{R} \rightarrow \mathbb{R}\). Decide whether this function is injective and whether it is surjective. What if it had been defined as \(\cos : \mathbb{R} \rightarrow[-1,1] ?\)

There are four different functions \(f:\\{a, b\\} \rightarrow\\{0,1\\} .\) List them. Diagrams suffice.

This problem concerns functions \(f:\\{1,2,3,4,5,6,7,8\\} \rightarrow\\{0,1,2,3,4,5,6\\} .\) How many such functions have the property that \(\left|f^{-1}(\\{2\\})\right|=4 ?\)

Given a sphere \(S,\) a great circle of \(S\) is the intersection of \(S\) with a plane through its center. Every great circle divides \(S\) into two parts. A hemisphere is the union of the great circle and one of these two parts. Prove that if five points are placed arbitrarily on \(S,\) then there is a hemisphere that contains four of them.

Is the function \(\theta: \mathscr{P}(\mathbb{Z}) \rightarrow \mathscr{P}(\mathbb{Z})\) defined as \(\theta(X)=\bar{X}\) bijective? If so, find \(\theta^{-1}\).

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 \(Y \subseteq B\). Prove or disprove: \(f^{-1}\left(f\left(f^{-1}(Y)\right)\right)=f^{-1}(Y)\).

This question concerns functions \(f:\\{A, B, C, D, E, F, G\\} \rightarrow\\{1,2,3,4,5,6,7\\} .\) How many such functions are there? How many of these functions are injective? How many are surjective? How many are bijective?