FS24

Task 1

a)

1.) Be careful with “compute the number of subsets of the set $\mathcal{P}(...)$“. This means we have to take the power set again, as it lists all subsets!

6.) Give an explicit expression for an injective function $f:\mathbb{Z}\to \mathbb{N}$ such that $f(0)=11$.

Be careful with the domain, that is $\mathbb{N}$. We need to have a case distinction between $\ge 0$ and $<0$.

A correct answer is $f(x)=\{\begin{array}{ll}2x+11& \text{if }x\ge 0\text{ (maps to odd numbers)}\\ -2x+10& \text{if }x<0\text{ (maps to even numbers)}\end{array}$ Note the $-$ on the bottom that makes sure we are always $\ge 0$.

b) Prove/Disprove partial order

For partial order prove/disprove, always try to disprove first. The statement “for all but finitely many n” sounds confusing at first, but you need to just get to the bottom of such things.

c)

Correctly apply the definition you moron. This was easy points. We need $(\rho \circ \tau {)}^2\subseteq (\rho \circ \tau )$. Function composition is associative (see Lemma 3.7).

e)

Countability task.

Task 2

c)

Note that we need to find all solutions, thus we give our answer in the format: $\{104+210k|k\in \mathbb{Z}\}$.

Task 3

a)

6.) $G$ group and $x\in G$ s.t. $\text{ord}(x)=8$. What is the order of $(x^{10},x^{12})\in G\times G$?

We have $\text{lcm}(10,8)=40$ and $40=10\cdot 4$, thus $\text{ord}(x^{10})=4$. In the same way $\text{ord}(x^{12})=2$.

Therefore $\text{ord}((x^{10},x^{12}))=\text{lcm}(2,4)=4$. Indeed $(x^{10},x^{12}{)}^4=(x^{40},x^{48})=((x^8{)}^5,(x^8{)}^6)=((1,1))$.

d) Prove T subgroup

This is still hard for me, see inverses using function proof. Maybe do this again.