Tutorial Exercises

1

1. It is an $A\to B$ function.
2. It is not a function as there are multiple relations from one domain, $d$
3. It is an $A\to B$ function.

2

One person could be the father to many people, so it can’t be a function. A person may not have a father so the entire codomain isn’t mapped hence not onto.

3

1. $\text{succ}:\mathbb{N}\to \mathbb{N}$ where $\text{succ}(n)=n+1$

It is both one-to-one and onto.

2. $+:(\mathbb{N}\times \mathbb{N})\to \mathbb{N}$ where $+(n,m)=n+m$

It is not one-to-one as there could be multiple combinations of numbers resulting in the same value in the codomain.

It is onto as the entire codomain is mapped.

4

Let $f:N\to N$ where:

1. $\{(1,2),(2,3),(3,4)\}$
2. $f(x)=x+1$
3. //
4. $\{(1,2),(2,2)\}$

5

a. Describe $g\circ f$.

$$\begin{array}{rl}g\circ f& =g(f(x))\\ & =\{\begin{array}{ll}g(k-3)& \text{if }k>10\\ g(k^2)& \text{if }k\le 10\end{array}\\ & =\{\begin{array}{ll}k-3+1& \text{if }k>10\text{ and }k-3\text{ is odd}\\ k-3& \text{if }k>10\text{ and }k-3\text{ is even}\\ k^2+1& \text{if }k\le 10\text{ and }{k}^2\text{ is odd}\\ k^2& \text{if }k\le 10\text{ and }{k}^2\text{ is even}\end{array}\\ & =\{\begin{array}{ll}k-2& \text{if }k>10\text{ and }k\text{ is even}\\ k-3& \text{if }k>10\text{ and }k\text{ is odd}\\ k^2+1& \text{if }k\le 10\text{ and }k\text{ is odd}\\ k^2& \text{if }k\le 10\text{ and }k\text{ is even}\end{array}\end{array}$$

b. Compute values of $(g\circ f)(-8)$ and $(g\circ f)(11)$.

• $(g\circ f)(-8)=64$
• $(g\circ f)(11)=8$

6

Take $x=1$, $f(1)=0\text{ and }2$. Take $x=0\text{ or }2$, $f(0)=-1\text{ and }\text{1}$ and $f(2)=\text{1}\text{ and }3$.

$f^{-1}(x)=f(x)$

7

Let $f:\mathbb{N}\to \mathbb{Z}$:

$$f(x)=\{\begin{array}{ll}\frac{x}{2}& \text{if }x\text{ is even}\\ -\frac{x+1}{2}& \text{if }x\text{ is odd}\end{array}$$

Hence:

$$\begin{array}{rl}f(0)& =0\\ f(1)& =-1\\ f(2)& =1\\ f(3)& =-2\\ f(4)& =2\\ & [..]\end{array}$$