#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)