Proof Practical Class, Week 22

Question One §

Question §

For which of the following pairs $A,B\subset \mathbb{R}$ does the relation $f=\{(x,y):x=\mid y|\}\subset A\times B$ define a mapping $f:A\to B$? Give reasons to your answers:

1. $A=\mathbb{R},B=[0,\infty )$.
2. $A=[0,\infty ),B=\mathbb{R}$.
3. $A=[0,\infty ),B=(-\infty ,0]$.

Solution §

…

Question Two §

Question §

Given the following mappings, determine which of the composites $f\circ g,g\circ f$ is defined, and write the resulting mapping in standard form:

$$\begin{array}{c}f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{R},f((x,y))=x+y,\text{and}\\ g:\mathbb{R}\to \mathbb{R},g(a)=a^2+1\end{array}$$

Solution §

…

Question Three §

Question §

For each of the following mappings, determine whether it is injective and/or surjective:

1. $f:\mathbb{Z}\to \mathbb{N},f(k)=k^2+2$.
2. $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R},f((x,y))=x+y$.
3. $f:\mathbb{N}\times \mathbb{N}\to \mathbb{N},f((m,n))=2^{m-1}\cdot (2n-1)$.

Solution §

…

Question Four §

Question §

Let $A=\mathcal{P}(\{u,v,w\})$ be the set of all subsets of $\{u,v,w\}$ and let $f:A\to A,f(x)=X\cap \{u,v\}$. Draw the diagram of the Cartesian product $A\times A$ and indicate $f$ as a subset of $A\times A$. What is the image of $f$?

Solution §

…

Question Five §

Question §

Find the image of the mapping $f:\mathbb{R}\to \mathbb{R},f(x)=\frac{1}{x^2+5}$.

Solution §

…

Question Six §

Question §

Show that $f:[3,\infty )\to (0,1],f(x)=\frac{2}{x-1}$ is a bijection and find the inverse mapping $f^{-1}$ - indicating its domain and image.

Solution §

…

Question Seven §

Question §

Let $f:\mathbb{R}\to \mathbb{R},f(x)=2x+1$. Define recursively the mappings $F_1=f$ and $F_{k+1}=f\circ F_k$ for all $k\in \mathbb{N}$. Compute several first mappings, conjecture an expression for $F_n$, and prove it by induction.

Solution §

…