Question

I guess this is something easy that I’m overlooking is a clear sign of illiteracy, but anyway.

How is that

(Map[Sign, LessEqual[x, y]]) === LessEqual[Sign[x], Sign[y]]
-> True  

But

(Map[Sign, LessEqual[-1, -100]]) == LessEqual[Sign[-1], Sign[-100]]
-> False  

Answer 1

Using Trace on the lhs will help to show what has happened.

Trace[Map[Sign, LessEqual[-1, -100]]]

Out[2]= {{-1 <= -100, False}, Sign /@ False, False}

Notice that Map has no HoldXXX attributes.

Attributes[Map]

Out[3]= {Protected}

So the LessEqual evaluates before Map does anything. At which point you get

Map[Sign,False]

As False is an atomic expression, this just evaluates to False.

The rhs of course evaluates to True, since Sign[-1] and Sign[-100] are both -1.

Daniel Lichtblau
Wolfram Research