Dictionaries are onto functions

Arturo Erdély — Sun, 19 Jun 2022 20:53:25 +0000

Dictionaries in Julia (and other programming languages) are onto functions, mathematically speaking. That is to say

f:A→Bf:A\rightarrow B

where

f

is a function defined for every

x∈A,x\in A,

and also for every

y∈By\in B

there exists at least one

x∈Ax\in A

such that

f (x) = y,

that is an onto (or surjective) function. The inverse image of

f

is a function

f(−1):℘(B)→℘(A)f^{(-1)}:\wp(B)\rightarrow\wp(A)

where

℘(B)\wp(B)

is the collection of all the subsets of

B,

that is the power set of

B,

and where

f(−1)(C)={x∈A:f(x)∈C}f^{(-1)}(C) = \{x \in A: f(x) \in C\}

for any

\subseteq B.

In other words, if

f

is a function that maps the elements of a set

A

onto a set

B

and if

C

is a subset of

B

then the inverse image

f^{(-1)}(C)

is a subset of

A

with all the elements that are mapped by

f

onto the set

C .

If D is a Dict then D is the equivalent to $f$ where $A =$ keys(D) and $B =$ values(D). To define the inverse image function $f^{(-1)}$ it is just necesary to code:

invImg(C::Set, D::Dict) = Set(filter(x -> D[x] ∈ C, keys(D)))

For example:

D = Dict([1,2,3,4,5,6,7] .=> ['a','a','b','a','b','a','c'])
C = Set(['a','c'])
invImg(C, D)

obtaining $f^{(-1)}(C) = \{1,2,4,6,7\}$ as expected.

Julia Community 🟣: Arturo Erdély

Dictionaries are onto functions