How to get Mathematica to stream a 2-variable function over two lists using functional programming methods?

Question

Suppose I have a function f[x_, y_] and two lists l1 , l2 . I would like to evaluate f[x,y] for each pair of x,y with x in l1 and y in l2 , and I would like to do this without doing all pairs of the form {l1[[i]],l2[[j]]} .

Essentially, I want something like Map [Map [f [# 1, # 2] &, l1], l2], where # 1 takes values ​​from l1 and # 2 takes values ​​from l2, but that doesn’t work .

(Motivation: I'm trying to implement some basic Haskell programs in Mathematica. In particular, I would like to be able to code the Haskell program

 isMatroid::[[Int]]->Bool isMatroid b =and[or[sort(union(xs\\[x])[y]'elem'b|y<-ys]|xs<-b,ys<-b, xs<-x] 

I think I can do the rest if I can figure out the original question, but I would like the code to look like Haskell. Any suggestions for implementing Haskell-like code in Mathematica would be appreciated.)

Answer 1

To evaluate the function f for all pairs from the two lists l1 and l2 , use Outer :

 In[1]:= Outer[f, {a,b}, {x,y,z}] Out[1]:= {{f[a,x],f[a,y],f[a,z]}, {f[b,x],f[b,y],f[b,z]}} 

Outer by default, runs at the lowest level of lists provided; you can also specify the level with an additional argument:

 In[2]:= Outer[f, {{1, 2}, {3, 4}}, {{a, b}, {c, d}}, 1] Out[2]:= {{f[{1,2},{a,b}], f[{1,2},{c,d}]}, {f[{3,4},{a,b}], f[{3,4},{c,d}]}} 

Note that this creates a nested list; You can Flatten if you want.

My original answer pointed to Thread and MapThread , which are two ways to apply a function to the corresponding pairs from lists, for example MapThread[f,{a,b},{1,2}] == {f[a,1], f[b,2]} .

PS I think that by studying these things you will find the documentation very useful. There are many common topic pages, such as applying functions to lists and manipulating a list . They are usually linked in the “more” section at the bottom of specific documentation. This greatly facilitates the search for things when you do not know what they will be called.

Answer 2

Raise an OP query for suggestions for implementing Haskell type code in Mathematica. A few things you will have to deal with:

• Haskell evaluates lazily; by default, Mathematica does not; he is very thirsty. You will need to struggle with Hold [] and its relatives to write lazily evaluating functions, but this can be done. You can also undermine the Mathematica evaluation process and mess with Prolog and Epilog, etc.
• The Haskell type and type system is probably more stringent than the default values ​​for Mathematica, but Mathematica has functions to implement strong type checking.

I am sure there is much more, but I am not very familiar with Haskell.

Answer 3

In [1]: = list1 = Range [1, 5];

In [2]: = list2 = Range [6, 10];

In [3]: = (f @@ #) and / @ Transpose [{list1, list2}]

Out [3] = {f [1, 6], f [2, 7], f [3, 8], f [4, 9], f [5, 10]}