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