7S Chapter 5 Exercises

exercise

Exercises from 7 Sketches, Chapter 5. Solutions: 7S Chapter 5 Solutions. Index: Map of Content.

Exercise 5.5

In the Prop $\mathbf{FinSet}$: 1. draw $f:3\to 2$ and $g:2\to 4$; 2. draw $f+g$; 3. composition rule; 4. identities; 5. the symmetry ${\sigma }_{m,n}$.

Sources: 7 Sketches, Exercise 5.5 and Solution A.5.

Solution: Solution 5.5

Exercise 5.9

Give three posetal props: posets $(\mathbb{N},⪯)$ with $m_1⪯n_1$, $m_2⪯n_2\Rightarrow m_1+m_2⪯n_1+n_2$.

Sources: 7 Sketches, Exercise 5.9 and Solution A.5.

Solution: Solution 5.9

Exercise 5.10

Spell out the five constituents of a Prop for $\mathbf{Bij}$, $\mathbf{Corel}$ or $\mathbf{Rel}$.

Sources: 7 Sketches, Exercise 5.10 and Solution A.5.

Solution: Solution 5.10

Exercise 5.16

Describe Port Graph composition visually and give an example.

Sources: 7 Sketches, Exercise 5.16 and Solution A.5.

Solution: Solution 5.16

Exercise 5.18

Draw the monoidal product of the port graph (5.15) with itself.

Sources: 7 Sketches, Exercise 5.18 and Solution A.5.

Solution: Solution 5.18

Exercise 5.20

Let $(P,{\le }_P)$ be the reflexive transitive closure of $R\subseteq P\times P$, $(Q,{\le }_Q)$ a preorder, $f:P\to Q$ a function. 1. If $R(x,y)\Rightarrow f(x)\le f(y)$, show $f$ is monotone. 2. Conversely, if $f$ is monotone then $R(x,y)\Rightarrow f(x)\le f(y)$.

Sources: 7 Sketches, Exercise 5.20 and Solution A.5.

Solution: Solution 5.20

Exercise 5.21

With $P,Q,R$ as in 5.20 and $g:Q\to P$: 1. If $a\le b\Rightarrow (g(a),g(b))\in R$, is $g$ monotone? 2. If $g$ is monotone, does $a\le b\Rightarrow (g(a),g(b))\in R$?

Sources: 7 Sketches, Exercise 5.21 and Solution A.5.

Solution: Solution 5.21

Exercise 5.23

Let $\mathcal{G}=\mathrm{Free}(G)$ for a graph $G=(V,A,s,t)$ and $\mathcal{C}$ a category. 1. Interpret $\mathrm{dom},\mathrm{cod}:\mathrm{Mor}(\mathcal{C})\to \mathrm{Ob}(\mathcal{C})$. 2. Show functors $\mathcal{G}\to \mathcal{C}$ correspond to pairs $(f:V\to \mathrm{Ob},g:A\to \mathrm{Mor})$ with $\mathrm{dom}(g(a))=f(s(a))$, $\mathrm{cod}(g(a))=f(t(a))$. 3. Is $(\mathrm{Mor}(\mathcal{C}),\mathrm{Ob}(\mathcal{C}),\mathrm{dom},\mathrm{cod})$ a graph? Use the word “adjunction”.

Sources: 7 Sketches, Exercise 5.23 and Solution A.5.

Solution: Solution 5.23

Exercise 5.24

The Free Monoid on $A$ is the free category on the one-vertex graph $\mathrm{Loop}(A)$. 1. Elements for $A=\{a\}$? 2. A well-known monoid isomorphic to it? 3. Elements for $A=\{a,b\}$?

Sources: 7 Sketches, Exercise 5.24 and Solution A.5.

Solution: Solution 5.24

Exercise 5.28

Show the Free Prop on the signature with one generator ${\rho }_{m,n}$ of every arity is the prop $\mathbf{PG}$ of port graphs.

Sources: 7 Sketches, Exercise 5.28 and Solution A.5.

Solution: Solution 5.28

Exercise 5.32

Draw $(f+{\mathrm{id}}_1+{\mathrm{id}}_1);(\sigma +{\mathrm{id}}_1);({\mathrm{id}}_1+h);\sigma ;g$ in the free prop on $f:1\to 1$, $g:2\to 2$, $h:2\to 1$.

Sources: 7 Sketches, Exercise 5.32 and Solution A.5.

Solution: Solution 5.32

Exercise 5.35

Is the Free Prop on $(G,s,t)$ the same as the prop presented by $(G,s,t,\varnothing )$?

Sources: 7 Sketches, Exercise 5.35 and Solution A.5.

Solution: Solution 5.35

Exercise 5.41

1. What is the multiplicative identity of the Rig ${\mathrm{Mat}}_n(R)$? 2. Show ${\mathrm{Mat}}_n(\mathbb{N})$ is not commutative.

Sources: 7 Sketches, Exercise 5.41 and Solution A.5.

Solution: Solution 5.41

Exercise 5.43

The Signal Flow Graph with inputs $x,y$ (copy $x$; amplify by 3; add; amplify by 5 and 3; …) produces which outputs?

Sources: 7 Sketches, Exercise 5.43 and Solution A.5.

Solution: Solution 5.43

Exercise 5.51

Direct sum of $A=\left(\begin{array}{ccc}3& 3& 1\\ 2& 0& 4\end{array}\right)$ and $B=(2561)$ in $\mathbf{Mat}(\mathbb{N})$.

Sources: 7 Sketches, Exercise 5.51 and Solution A.5.

Solution: Solution 5.51

Exercise 5.55

What matrices do the two signal flow graphs of Eq. (5.47) (copy then copy the top wire; copy then copy the bottom wire) represent? Are they equal?

Sources: 7 Sketches, Exercise 5.55 and Solution A.5.

Solution: Solution 5.55

Exercise 5.58

Draw signal flow graphs for 1. $\left(\begin{array}{c}0\\ 1\\ 2\end{array}\right)$; 2. $\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$; 3. $\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)$.

Sources: 7 Sketches, Exercise 5.58 and Solution A.5.

Solution: Solution 5.58

Exercise 5.59

Prove Proposition 5.56 in general: construct $g:m\to n$ with $S(g)=M$ as four layers.

Sources: 7 Sketches, Exercise 5.59 and Solution A.5.

Solution: Solution 5.59

Exercise 5.62

For each matrix of 7S Exercise 5.58 draw a different representing graph and prove equality using Theorem 5.60.

Sources: 7 Sketches, Exercise 5.62 and Solution A.5.

Solution: Solution 5.62

Exercise 5.63

For the two graphs of Eq. (5.64) over $\mathbb{N}$: 1. show without computing matrices that they differ; 2. over $\mathbb{N}/3\mathbb{N}$ find a minimal representation.

Sources: 7 Sketches, Exercise 5.63 and Solution A.5.

Solution: Solution 5.63

Exercise 5.67

Show $(\mathbb{R},\ast ,1)$ and $(\mathbb{R},+,0)$ are commutative monoid objects in $(\mathbf{Set},1,\times )$.

Sources: 7 Sketches, Exercise 5.67 and Solution A.5.

Solution: Solution 5.67

Exercise 5.69

$U:\mathbf{Mat}(R)\to \mathbf{Set}$, $n\mapsto R^n$, $M\mapsto (v\mapsto vM)$. 1. Show $U$ preserves unit and product. 2. Show it carries monoid objects to monoids. 3. Which monoid structure on $\mathbb{R}$ does $(1,\text{add},\text{zero})$ give?

Sources: 7 Sketches, Exercise 5.69 and Solution A.5.

Solution: Solution 5.69

Exercise 5.77

Behaviours of the reversed add icon $1\to 2$ and reversed copy icon $2\to 1$.

Sources: 7 Sketches, Exercise 5.77 and Solution A.5.

Solution: Solution 5.77

Exercise 5.80

Write $B+C$ for $B\subseteq R^m\times R^n$, $C\subseteq R^p\times R^q$ in ${\mathbf{Rel}}_R$.

Sources: 7 Sketches, Exercise 5.80 and Solution A.5.

Solution: Solution 5.80

Exercise 5.82

Show $B(g;h^{\mathrm{op}})=\{(x,y)\mid S(g)(x)=S(h)(y)\}$.

Sources: 7 Sketches, Exercise 5.82 and Solution A.5.

Solution: Solution 5.82

Exercise 5.83

Show $B(g^{\mathrm{op}};h)=\{(S(g)x,S(h)x)\mid x\in R^m\}$.

Sources: 7 Sketches, Exercise 5.83 and Solution A.5.

Solution: Solution 5.83

Exercise 5.84

Over a field $R$: 1. composing $g$ with reversed zeros gives $\ker S(g)$; 2. composing reversed discards with $g$ gives $\mathrm{im}S(g)$; 3. $B(g)$ is a linear subspace.

Sources: 7 Sketches, Exercise 5.84 and Solution A.5.

Solution: Solution 5.84

Exercise 5.85

Show the composite of linear relations $B\subseteq R^m\times R^n$, $C\subseteq R^n\times R^p$ is linear.

Sources: 7 Sketches, Exercise 5.85 and Solution A.5.

Solution: Solution 5.85