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Exercise 48 - Chapter 4

edited August 2018

$$\def\cat#1{{\mathcal{#1}}}$$ $$\def\comp#1{{\widehat{#1}}}$$ $$\def\conj#1{{\check{#1}}}$$ $$\def\id{{\mathrm{id}}}$$ $$\def\true{{\mathrm{true}}}$$ $$\def\false{{\mathrm{false}}}$$ $$\def\Cat#1{{\textbf{#1}}}$$ $$\def\BB{{\mathbb{B}}}$$ $$\def\ZZ{{\mathbb{Z}}}$$ $$\def\tn#1{{\text{#1}}}$$ Consider the monoidal category $$(\Cat{Set},1,\times)$$, together with the diagram

Suppose that $$A=B=C=D=F=G=\ZZ$$ and $$E=\BB={\true,\false}$$, and suppose that $$f_C(a)=|a|$$, $$f_D(a)=a*5$$, $$g_E(d,b)=d\leq b$$, $$g_F(d,b)=d-b$$, and $$h(c,e)=\tn{if }e\tn{ then }c\tn{ else }1-c$$.

1. What are $$g_E(5,3)$$ and $$g_F(5,3)$$?
2. What are $$g_E(3,5)$$ and $$g_F(3,5)$$?
3. What is $$h(5,\true)$$?
4. What is $$h(-5,\true)$$?
5. What is $$h(-5,\false)$$?

The whole diagram now defines a function $$A\times B\to G\times F$$; call it $$q$$.

1. What are $$q_G(-2,3)$$ and $$q_F(-2,3)$$?
2. What are $$q_G(2,3)$$ and $$q_F(2,3)$$?

• Options
1.
1. $$g_E(5,3)=\mathrm{false}$$, $$g_F(5,3)=2$$.
2. $$g_E(3,5)=\mathrm{true}$$, $$f_F(3,5)=-2$$.
3. $$h(5,\mathrm{true})=5$$.
4. $$h(-5,\mathrm{true})=-5$$.
5. $$h(-5,\mathrm{false})=6$$.



1. $$q_G(-2,3)=2$$, $$q_F(-2,3)=-13$$.
2. $$q_G(2,3)=-1$$, $$q_F(2,3)=7$$.

In general, $$q_G(a,b)=\begin{cases}\lvert a\rvert\,\,\text{if}\,5a\leq b\\1-\lvert a\rvert\,\,\text{otherwise}\end{cases}$$ and $$q_F(a,b)=5a-b$$.

Comment Source:1. \$$g\_E(5,3)=\mathrm{false}\$$, \$$g\_F(5,3)=2\$$. 2. \$$g\_E(3,5)=\mathrm{true}\$$, \$$f\_F(3,5)=-2\$$. 3. \$$h(5,\mathrm{true})=5\$$. 4. \$$h(-5,\mathrm{true})=-5\$$. 5. \$$h(-5,\mathrm{false})=6\$$. \$$\$$ 1. \$$q\_G(-2,3)=2\$$, \$$q\_F(-2,3)=-13\$$. 2. \$$q\_G(2,3)=-1\$$, \$$q\_F(2,3)=7\$$. In general, \$$q\_G(a,b)=\begin{cases}\lvert a\rvert\,\,\text{if}\,5a\leq b\\\\1-\lvert a\rvert\,\,\text{otherwise}\end{cases}\$$ and \$$q\_F(a,b)=5a-b\$$.