292 lines
6.3 KiB
Typst
292 lines
6.3 KiB
Typst
|
#import "../handout.typ": *
|
||
|
|
#import "../macros.typ": *
|
||
|
|
|
||
|
|
= Tropical Arithmetic
|
||
|
|
|
||
|
|
#definition()
|
||
|
|
The _tropical sum_ of two numbers is their minimum:
|
||
|
|
$
|
||
|
|
x #tp y = min(x, y)
|
||
|
|
$
|
||
|
|
|
||
|
|
#definition()
|
||
|
|
The _tropical product_ of two numbers is their sum:
|
||
|
|
$
|
||
|
|
x #tm y = x + y
|
||
|
|
$
|
||
|
|
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
- Is tropical addition commutative? \
|
||
|
|
#note([i.e, does $x #tp y = y #tp x$?])
|
||
|
|
- Is tropical addition associative? \
|
||
|
|
#note([i.e, does $(x #tp y) #tp z = x #tp (y #tp z)$?])
|
||
|
|
- Is there a tropical additive identity? \
|
||
|
|
#note([i.e, is there an $i$ so that $x #tp i = x$ for all real $x$?])
|
||
|
|
|
||
|
|
#solution([
|
||
|
|
- Is tropical addition commutative?\
|
||
|
|
Yes, $min(min(x,y),z) = min(x,y,z) = min(x,min(y,z))$
|
||
|
|
- Is tropical addition associative? \
|
||
|
|
Yes, $min(x,y) = min(y,x)$
|
||
|
|
- Is there a tropical additive identity? \
|
||
|
|
No. There is no $n$ where $x <= n$ for all real $x$
|
||
|
|
])
|
||
|
|
|
||
|
|
#v(1fr)
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Let's expand $#sym.RR$ to include a tropical additive identity.
|
||
|
|
- What would be an appropriate name for this new number?
|
||
|
|
- Give a reasonable definition for...
|
||
|
|
- the tropical sum of this number and a real number $x$
|
||
|
|
- the tropical sum of this number and itself
|
||
|
|
- the tropical product of this number and a real number $x$
|
||
|
|
- the tropical product of this number and itself
|
||
|
|
|
||
|
|
#solution([
|
||
|
|
#sym.infinity makes sense, with
|
||
|
|
$#sym.infinity #tp x = x$; #h(1em)
|
||
|
|
$#sym.infinity #tp #sym.infinity = #sym.infinity$; #h(1em)
|
||
|
|
$#sym.infinity #tm x = #sym.infinity$; #h(1em) and
|
||
|
|
$#sym.infinity #tm #sym.infinity = #sym.infinity$
|
||
|
|
])
|
||
|
|
|
||
|
|
#v(1fr)
|
||
|
|
#pagebreak() // MARK: page
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Do tropical additive inverses exist? \
|
||
|
|
#note([Is there an inverse $y$ for every $x$ so that $x #tp y = #sym.infinity$?])
|
||
|
|
|
||
|
|
#solution([
|
||
|
|
No. Unless $x = #sym.infinity$, there is no x where $min(x, y) = #sym.infinity$
|
||
|
|
])
|
||
|
|
|
||
|
|
#v(1fr)
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Is tropical multiplication associative? \
|
||
|
|
#note([Does $(x #tm y) #tm z = x #tm (y #tm z)$ for all $x,y,z$?])
|
||
|
|
|
||
|
|
#solution([Yes, since (normal) addition is associative])
|
||
|
|
|
||
|
|
#v(1fr)
|
||
|
|
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Is tropical multiplication commutative? \
|
||
|
|
#note([Does $x #tm y = y #tm x$ for all $x, y$?])
|
||
|
|
|
||
|
|
#solution([Yes, since (normal) addition is commutative])
|
||
|
|
|
||
|
|
#v(1fr)
|
||
|
|
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Is there a tropical multiplicative identity? \
|
||
|
|
#note([Is there an $i$ so that $x #tm i = x$ for all $x$?])
|
||
|
|
|
||
|
|
#solution([Yes, it is 0.])
|
||
|
|
|
||
|
|
#v(1fr)
|
||
|
|
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Do tropical multiplicative inverses always exist? \
|
||
|
|
#note([
|
||
|
|
For every $x != #sym.infinity$, does there exist an inverse $y$ so that $x #tm y = i$, \
|
||
|
|
where $i$ is the additive identity?
|
||
|
|
])
|
||
|
|
|
||
|
|
#solution([Yes, it is $-x$. For $x != 0$, $x #tm (-x) = 0$])
|
||
|
|
|
||
|
|
#v(1fr)
|
||
|
|
#pagebreak() // MARK: page
|
||
|
|
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Is tropical multiplication distributive over addition? \
|
||
|
|
#note([Does $x #tm (y #tp z) = x #tm y #tp x #tm z$?])
|
||
|
|
|
||
|
|
#solution([Yes, $x + min(y,z) = min(x+y, x+z)$])
|
||
|
|
|
||
|
|
#v(1fr)
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Fill the following tropical addition and multiplication tables
|
||
|
|
|
||
|
|
#let col = 10mm
|
||
|
|
|
||
|
|
#notsolution(
|
||
|
|
table(
|
||
|
|
columns: (1fr, 1fr),
|
||
|
|
align: center,
|
||
|
|
stroke: none,
|
||
|
|
table(
|
||
|
|
columns: (col, col, col, col, col, col),
|
||
|
|
align: center,
|
||
|
|
table.header(
|
||
|
|
[$#tp$],
|
||
|
|
[$1$],
|
||
|
|
[$2$],
|
||
|
|
[$3$],
|
||
|
|
[$4$],
|
||
|
|
[$#sym.infinity$],
|
||
|
|
),
|
||
|
|
|
||
|
|
box(inset: 3pt, $1$), [], [], [], [], [],
|
||
|
|
box(inset: 3pt, $2$), [], [], [], [], [],
|
||
|
|
box(inset: 3pt, $3$), [], [], [], [], [],
|
||
|
|
box(inset: 3pt, $4$), [], [], [], [], [],
|
||
|
|
box(inset: 3pt, $#sym.infinity$), [], [], [], [], [],
|
||
|
|
),
|
||
|
|
table(
|
||
|
|
columns: (col, col, col, col, col, col),
|
||
|
|
align: center,
|
||
|
|
table.header(
|
||
|
|
[$#tm$],
|
||
|
|
[$0$],
|
||
|
|
[$1$],
|
||
|
|
[$2$],
|
||
|
|
[$3$],
|
||
|
|
[$4$],
|
||
|
|
),
|
||
|
|
|
||
|
|
box(inset: 3pt, $0$), [], [], [], [], [],
|
||
|
|
box(inset: 3pt, $1$), [], [], [], [], [],
|
||
|
|
box(inset: 3pt, $2$), [], [], [], [], [],
|
||
|
|
box(inset: 3pt, $3$), [], [], [], [], [],
|
||
|
|
box(inset: 3pt, $4$), [], [], [], [], [],
|
||
|
|
),
|
||
|
|
),
|
||
|
|
)
|
||
|
|
|
||
|
|
#solution(
|
||
|
|
table(
|
||
|
|
columns: (1fr, 1fr),
|
||
|
|
align: center,
|
||
|
|
stroke: none,
|
||
|
|
table(
|
||
|
|
columns: (col, col, col, col, col, col),
|
||
|
|
align: center,
|
||
|
|
table.header(
|
||
|
|
[$#tp$],
|
||
|
|
[$1$],
|
||
|
|
[$2$],
|
||
|
|
[$3$],
|
||
|
|
[$4$],
|
||
|
|
[$#sym.infinity$],
|
||
|
|
),
|
||
|
|
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
|
||
|
|
box(inset: 3pt, $#sym.infinity$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
box(inset: 3pt, $#sym.infinity$),
|
||
|
|
),
|
||
|
|
table(
|
||
|
|
columns: (col, col, col, col, col, col),
|
||
|
|
align: center,
|
||
|
|
table.header(
|
||
|
|
[$#tm$],
|
||
|
|
[$0$],
|
||
|
|
[$1$],
|
||
|
|
[$2$],
|
||
|
|
[$3$],
|
||
|
|
[$4$],
|
||
|
|
),
|
||
|
|
|
||
|
|
box(inset: 3pt, $0$),
|
||
|
|
box(inset: 3pt, $0$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $1$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
box(inset: 3pt, $5$),
|
||
|
|
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $2$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
box(inset: 3pt, $5$),
|
||
|
|
box(inset: 3pt, $6$),
|
||
|
|
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $3$),
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
box(inset: 3pt, $5$),
|
||
|
|
box(inset: 3pt, $6$),
|
||
|
|
box(inset: 3pt, $7$),
|
||
|
|
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
box(inset: 3pt, $4$),
|
||
|
|
box(inset: 3pt, $5$),
|
||
|
|
box(inset: 3pt, $6$),
|
||
|
|
box(inset: 3pt, $7$),
|
||
|
|
box(inset: 3pt, $8$),
|
||
|
|
),
|
||
|
|
),
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
#v(2mm)
|
||
|
|
|
||
|
|
#problem()
|
||
|
|
Expand and simplify $f(x) = (x #tp 2)(x #tp 3)$, then evaluate $f(1)$ and $f(4)$ \
|
||
|
|
#hint([Adjacent parenthesis imply tropical multiplication])
|
||
|
|
|
||
|
|
#solution([
|
||
|
|
$
|
||
|
|
(x #tp 2)(x #tp 3)
|
||
|
|
&= x^2 #tp 2x #tp 3x #tp (2 #tm 3) \
|
||
|
|
&= x^2 #tp (2 #tp 3)x #tp (2 #tm 3) \
|
||
|
|
&= x^2 #tp 2x #tp 5
|
||
|
|
$
|
||
|
|
|
||
|
|
Also, $f(1) = 2$ and $f(4) = 5$.
|
||
|
|
])
|
||
|
|
|
||
|
|
#v(1fr)
|