How can I substitute something using only map?

Question

For example, I have the equation

x^2 + y^2 == x*y

and I want to apply the rule

y -> s*x

I can do it easily by

x^2 + y^2 == x*y //. y -> s*x

But it seems to me it would be possible to use only Map to change everything. How can I do it that way? In other words, I think Replace is just a kind of syntactic sugar, and want to know how to express Replace by Map.

"I think Replace is just a kind of syntactic sugar" - no. No, it's not. — J. M.'s missing motivation, May 16 '13 at 02:00
@J.M. Agree. But perhaps it could be interesting if the OP tells us why he thinks it is — Dr. belisarius, May 16 '13 at 02:04
I think it would be possible if there is a "function" that changes y to s*x. It seems rules are not the function...... — poorGuy, May 16 '13 at 02:09
Yes, they act quite differently; Map[f, {1, 2}] will apply the function f on the elements of the list {1, 2} to yield {f[1], f[2]}. That wors differently from Replace[{1, 2}, 1 -> 5, {1}] which replaces the 1 in {1, 2} with 5, to yield {5, 2}. — J. M.'s missing motivation, May 16 '13 at 02:22
f[y]=s x; f[v_]:=v; Print[":("]; Map[f,x^2+y^2,Infinity] Might want to check out DownValues[f] to see that this in fact does not avoid rules. — ssch, May 16 '13 at 02:24

score 5 · Answer 1 · edited Apr 13 '17 at 12:55

5

As expressed in the comments, the Replace functions are not merely "syntactic sugar" for Map. The two are quite different. One primary difference is the order in which expressions are visited. See:
How to perform a depth-first preorder traversal of an expression? Another is that Replace will go inside held expressions, while Map does not evaluate:

Hold[1 + 2 + 3] /. {2 -> 7}

Hold[1 + 7 + 3]

Map[Print, Hold[1 + 2 + 3], {2}]

Hold[Print[1] + Print[2] + Print[3]]

With the exception of these important differences, one can do something resembling a ReplaceAll using MapAll, like this:

SetAttributes[f, HoldAll]

f[y] := s*x
f[other_] := other

f //@ (x^2+y^2==x*y)

x^2 + s^2 x^2 == s x^2

edited Apr 13 '17 at 12:55

Community

1

answered May 16 '13 at 02:34

Mr.Wizard

271,378
34
587
1,371

Nevertheless, the last use of //@ just feels like using a pipe wrench to whack nails... – J. M.'s missing motivation May 16 '13 at 03:22
Oh god Thank you! – poorGuy May 16 '13 at 03:29

How can I substitute something using only map?

1 Answers1