Defining a general function f with slot-holder arguments #1, #2

Question

I am trying to get a general function $f$ accept slot-holder arguments #1, #2, where the actual arguments are lists {a1, a2,...} and {b1, b2,....}. I thought that this would involve some combination of the functions "Map" and "Apply" - I have experimented with each function individually (I don't know how to combine them) but so far I have not managed to get the desired results.

1. I work only with "Apply", and this has no problem when the passed-in arguments are just single variables (not lists):

   Apply[f[#1, #2] &, {0, 1}]

The outcome is: f[0, 1]

When I try to pass in lists, for example #1 should accept {0,2} and #2 should accept {1,2} to produce a list {f[0,1], f[2,2]}, the output is instead:

f[{0, 2}, {1, 2}]

The code I used is:

   Apply[f[#1, #2] &, {{0, 2}, {1, 2}}]

However, I noticed that if the function is explicitly defined, for example take a simple function like f[#1, #2]=#1+#2, then the output is correct.

Apply[#1 + #2 &, {{0, 2}, {1, 2}}]

which produces:

{1, 4}

2. I also try to use "Map" (or /@), but it only works with a single slot-holder argument. For example

   f[#] & /@ {a, b, c, d, e}

gives:

{f[a], f[b], f[c], f[d], f[e]}

but if I put:

f[#1, #2] & /@ {{a, b, c, d, e}, {m, n, p, q, r}}

the result is:

{f[{a, b, c, d, e}, #2], f[{m, n, p, q, r}, #2]}

and not the desired result, which should be:

{f[a,m], f[b,n], f[c,p], f[d,q], f[e,r]}

I would be grateful if anyone could help me with this.

Answer 1

Apply[f,arg] (or f /@ arg)is the same as f[arg]. If the function needs 2 arguments, you need to deliver 2 arguments. E.g.

(#1 + #2) &[{1, 2}, {3, 4}]
(* {4, 6} *)

Equally, the larger form needs a list of 2 arguments:

Apply[(#1 + #2) &, {{1, 2}, {3, 4}}]
(* {4, 6} *)

Map , on the other hand, applies the function to every element of the argument t. By default to the first level of the argument. E.g.:

Map[(# + 1) &, {1, 2}]
(* {2, 3} *)

or

(# + 1) & /@ {1, 2}
(* {2, 3} *)