Why False && {False, False} does not preserve list structure?

Question

I am on Mathematica 12.1.1.0 trying to perform a boolean operation over every element of a list, but preserving the list structure.

When I run

True && {True, True}

the result is

{True, True}

However, if I run

False && {False, False}

the result is

False

i.e., the list structure disappears.

What is the reason? And what is the best way to combine a single boolean condition with every element of a list, preserving the list structure?

Answer 1

Welcome to MMA SE! The reason is that And (&&) is short-circuited. This means that as soon as an argument is False (starting from the left), the whole thing immediately evaluates to False. So False && expr is False for any expr.

The opposite behavior occurs when the first argument is True: True && expr gives expr for any expr. If the first argument is True, then you still need to check the next arguments. But you know that whatever the output of And is, it will be given by those next arguments; so you can disregard the initial True. And from the docs:

And gives symbolic results when necessary, removing initial arguments that are True.

So && is not actually preserving the list structure in the first case; it's just returning the second argument.

To preserve the list structure, one solution is to use Map to map a function over a list, for which there is the syntactic sugar /@. We can map the anonymous functions True && # & and False && # & over the list:

True && # & /@ {False, True}

False && # & /@ {False, True}

However, depending on your application, you might want to do things slightly differently. Defining a symbol to "mean" And but have the attribute Listable comes to mind, for example (since the attribute gets applied before the definition does):

myAnd[args___] := And[args]
SetAttributes[myAnd, Listable]

myAnd[True, {False, True}]
myAnd[False, {False, True}]

The Help docs for Function and Map likely link to other things you might find useful at the bottom!